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IN  MEMORIAM 
FLORIAN  CAJORl 


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i-i-t.^        (—-^y^^X^L^ 


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in  2007  with  funding  from 

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THE  ELEMENTS  OF 

PLANE  AND  SPHERICAL 
TRIGONOMETRY 


•y^)^' 


THE  ELEMENTS  OF 
PLANE  AND  SPHERICAL 

TEIGONOMETRY 


BY 

JOHN  GALE   HUN 

n 

AND 

CHARLES  RANALD  MacINNES 


J .  > »  » 


Neto  ¥otfe 
THE  MACMILLAN  COMPANY 

LONDON:  MACMILLAN  &  CO.,  Ltd. 
1911 


Copyright,  1911 
By  the  MACMILLAN  COMPANY 


Set  up,  electrotyped  and  printed  Aug.  1911 


CAJORI 


Press  of 
The  new  Era  Printing  company 

LANCASTER.   PA. 


PREFACE. 

The  subject  matter  of  the  chapters  of  this  book  devoted  to 
spherical  trigonometry  has  been  used  in  pamphlet  form  for 
the  last  four  years  in  Princeton  University,  and  that  of  the 
chapters  on  plane  trigonometry  for  the  last  three  years. 

The  aim  of  the  authors  has  been  to  present  in  as  brief  and 
clear  a  manner  as  possible  the  essentials  of  a  short  course  in 
trigonometry.  It  has  been  found  that  the  plane  trigonometry 
may  be  covered  in  about  thirty  recitations,  and  the  spherical 
trigonometry  in  somewhat  less  than  this  time. 

It  has  been  thought  advisable  to  devote  some  time  to  drawing 
the  graphs  of  simple  equations  in  polar  coordinates.  The  reason 
for  this  is  two-fold.  Firstly,  because  such  problems  aid  in 
giving  the  student  a  clearer  idea  of  the  way  in  which  the  trigo- 
nometric functions  vary  as  the  angle  is  changed ;  and  secondly, 
because  of  a  very  common  lack  of  sufficient  knowledge  of  polar 
coordinates  on  the  part  of  students  beginning  the  study  of 
calculus. 

The  fact  that  the  trigonometric  functions  are  ratios  of  line 
segments  is  emphasized,  and  their  representation  by  means  of 
lengths  of  lines  is  used  as  little  as  is  conveniently  possible. 

Certain  of  the  proofs  of  theorems  are  shorter  than  in  many 
text  books,  and,  it  is  hoped,  thereby  made  more  clear;  notably 
the  proofs  of  the  formulae  for  sin  (A  =1=  B)  and  cos  (A  =*=  B). 

The  logarithmic  tables  were  taken  from  Crockett's  five  place 
tables,  and  the  proof  sheets  carefully  compared  with  Albrecht's 
tables.  It  is  hoped  that  few  errors  will  be  found.  The  explana- 
tion of  the  tables  will  be  found  at  the  end  of  the  book.  It  is 
designed  that  the  matter  there  contained  be  given  as  lessons, 
and  for  this  reason  the  student  is  given  examples  involving  the 
difficulties  usually  encountered  by  the  beginner. 

The  authors  wish  to  thank  the  members  of  the  mathematical 
department  of  Princeton  University  who  have  kindly  suggested 
several  changes  from  the  form  used  in  the  pamphlet  editions. 
Princeton,  N.  J., 
March  1,  1911. 

918239 


CONTENTS. 


Chapter  I 1 

Chapter  II 23 

Chapter  III.     Trigonometric  Analysis 30 

Chapter  IV 53 

Chapter  V.     Introductory  Review 66 

Chapter  VI.     Relations  between  the  Sides  and  Angles 

of  a  Spherical  Triangle 73 

Chapter  VII.  The  Right  Spherical  Triangle.  ...  80 
Chapter  VIII.  The  Solution  of  Oblique  Triangles.  .  86 
Chapter  IX.     Other    Formulae    Relating   to    Spherical 

Triangles 98 

Table  I.  Logarithms  of  Numbers  from  1  to  10,000  .  103 
Table  II.  Logarithms  of  the  Trigonometric  Functions.  123 
Table  III.  Natural  Trigonometric  Functions.  .  .  .169 
Explanation  of  Tables 193 


vu 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


CHAPTER  I. 


1.  Positive  and  Negative  Quantities.  It  is  of  fundamental 
importance  for  the  student  of  trigonometry  to  realize  at  the 
outset  the  meaning  of  negative  as  well  as  of  positive  quantities. 
That  a  negative  result  may  have  an  actual  meaning  is  shown 
by  the  following  examples :  Let  us  suppose  that  a  man  has  x 
dollars  and  that  he  owes  y  dollars.  Then  the  difference  x  —  y 
denotes  the  number  of  dollars  he  will  have  after  paying  his 
debts.  If  now  this  difference  happens  to  be  negative,  it  will 
indicate  the  number  of  dollars  he  will  still  owe  after  he  has  paid 
out  all  of  his  cash. 

As  a  second  example  consider  the  following:  A  man  A  is 
observed  to  pass  a  certain  inn  at  exactly  noon,  walking  at  the 
rate  of  a  miles  per  hour.  At  two  o'clock  another  man  B  passes 
the  same  point  at  the  rate  of  h  miles  per  hour.  How  far  will 
they  be  beyond  the  inn  when  B  overtakes  A? 

Let  the  required  distance  be  denoted  by  x.  A  will  take  x/a 
hours  to  Yv^alk  the  distance,  and  B  will  take  xjh  hours.  Also 
since  A  has  a  two-hour  start  on  J5,  the  latter  must  walk  the  x 
miles  in  two  hours  less  than  A  requires.  We  therefore  have 
the  equation 

-  =  -  -  2 
h       a 

and  hence 

2ah 
X  = 


b  —  a 


Now  suppose  that  A  walks  two  miles  and  B  four  miles  per 
hour.  Then  B  will  overtake  A  2.2.4/(4—  2)  or  eight  miles 
beyond  the  inn.  If  on  the  other  hand  A  walks  three  miles 
and  B  two  miles  per  hour,  they  will  be  together  2.3.2/(2  —  3) 
or  —  12  miles  beyond  the  inn.  The  interpretation  of  this 
1  1 


A 

B 

C 

c* 

s> 

B 

o 

2  ELEMENTS  OF  PLANE   TRIGONOMETRY. 

negative  result  is  evidently  that  they  were  together  twelve 
miles  before  they  came  to  the  inn.  That  this  is  true  is  easily 
verified  as  follows:  Suppose  they  are  together  at  this  place. 
It  will  take  A  12/3  or  four  hours  to  reach  the  inn,  and  B  12/2 
or  six  hours.  Hence,  if  A  gets  there  at  noon,  B  will  arrive 
at  two  o'clock  as  given  in  the  problem. 

2.  Positive  and  Negative  Lines.  The  second  problem  leads 
us  naturally  to  the  idea  of  positive  and  negative  directions 
along  a  fine;  that  is,  it  brought  out  the  fact  that  if  we  assume 

one  direction  of  a  line  to 
be  positive,  the  opposite 
direction  algebraically  is 
jTjQ   I  negative.     Let  A  and  B 

be  two  points  of  a  line. 
By  the  symbol  AB  we  shall  mean  not  only  the  distance 
between  A  and  B,  but  also  the  direction  from  A  to  B. 
The  easiest  way  in  which  to  think  of  the  symbol  AB  is  to 
consider  it  as  expressing  a  motion  along  the  line  from  A  to 
B.  Let  us  now  have  a  third  point  C  on  the  line.  By  the 
expression  AB  -\-  BC  we  mean  a  motion  from  A  to  5  and  thence 
to  C.  The  total  motion  is  therefore  merely  from  A  to  C.  This 
is  expressed  by  the  equation  AB  +  BC  =  AC.  It  is  to  be 
carefully  observed  that  this  equation  is  true  no  matter  in  what 
order  the  points  A,  B,  and  C  may  occur  on  the  line.  In  par- 
ticular we  have  AB  +  BA  =  A  A  =  0,  or  AB  =  —  BA. 

3.  Positive  and  Negative  Angles.  Let  OX  be  a  fixed  line 
and  let  the  positive  direction  be  from  0  to  X.  Also  let  a 
second  line  OP,  whose  positive  direction  is  from  0  to  P,  rotate 
about  0  in  the  plane  of  the  paper.  The  two  lines  OX  and  OP 
are  said  to  form  the  angle  XOP.  Just  as  the  symbol  AB 
denotes  not  only  the  distance  between  A  and  B,  but  also  the 
direction  from  A  to  B,  so  the  symbol  XOP  denotes  not  only 
the  size  of  the  angle  formed  by  the  lines  OX  and  OP,  but  also 
the  direction  of  rotation  from  OX  into  OP.  We  may  consider 
XOP  as  denoting  a  turning  of  the  line  OX  through  a  given 
angle  in  a  given  direction.  Let  us  now  have  another  line  OQ 
through  0.  The  symbol  XOP  +  POQ  then  denotes  a  rotation 
of  OX  into  the  position  of  OP  and  thence  into  that  of  OQ. 


ELEMENTS   OF   PLANE   TRIGONOMETRY. 


>X 


The  total  rotation  is  merely  from  OX  into  OQ,  and  therefore 
we  have  the  equation  XOP  +  POQ  =  XOQ.  In  particular 
XOP  +  POX  =  XOX  =  0,  or  XOP  =  -  POX.  In  the  angle 
XOP,  OX  is  known  as  the  initial  line  and  OP  as  the  terminal 
line. 

We  may  choose  either  direction  of  rotation  as  positive.  It 
is  however  customary  to 
take  as  positive  the 
counter-clockwise  rota- 
tion, that  is,  a  rotation 
in  the  direction  opposite 
to  that  of  the  hands  of 
a  clock.  Thus  in  Fig.  2 
XOP,  XOQ,  and  POQ  are 
positive  angles,  while  POX,  QOX,  and  QOP  are  negative  angles. 
4.  Measurement  of  Angles.  Two  systems  of  measuring 
angles  are  in  common  use.  They  are  known  as  the  sexagesimal 
system  and  the  radian  or  circular  measure  system. 

In  the  sexagesimal  system  a  right  angle  is  divided  into  ninety 
equal  parts  called  degrees.  Each  degree  is  in  turn  divided 
into  sixty  equal  parts  called  minutes,  and  finally  each  minute 
is  divided  into  sixty  equal  parts  called  seconds.  To  avoid 
writing  degrees,  minutes  and  seconds,  the  symbols  °,  ',  "  are 
employed.  Thus  69°  34'  26''  is  read  sixty-nine  degrees,  thirty- 
four  minutes  and  twenty-six  sec- 
onds. 

In  the  radian  system  the  unit 
angle  is  that  angle  which  at  the 
centre  of  a  circle  subtends  an  arc 
equal  in  length  to  the  radius. 
This  unit  angle  is  known  as  a 
radian. 

In  Fig.  3  let  AOP  be  an  angle 

at  the  center  of  a  circle  of  radius 

If  this   angle  is  equal  to   a 


Fig.  3. 

radian,  we  have 
AOP 


r. 


circumference 


360°  =  arc  AP 
=  r  :  27rr, 
and  hence  AOP  =  one  radian  =  360°/27r  =  57.2°  -f. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


If  we  draw  a  circle  of  any  radius  r  with  the  centre  at  the 
vertex  of  a  given  angle,  this  angle  will  contain  as  many  radians 
as  the  arc  subtended  by  it  contains  r.  Hence  the  radian  measure 
of  an  angle  is  the  ratio  of  the  length  of  the  arc  to  the  radius. 
Thus  the  arc  subtended  by  360°  is  27rr,  and  therefore  the  radian 
measure  of  360°  is  27rr/r  or  27r.  Similarly  the  radian  measure 
of  90°  is  7r/2,  that  of  60°  is  7r/3,  etc. 

5.  Exercise  I. 

L  Express  each  of  the  following  angles  in  radians:  30°,  150°, 
750°,  45°,  260°,  -  75°. 

2.  Express  each  of  the  following  angles  in  degrees,  minutes, 
and  seconds  f  tt,  ^-tt,  yx,  f  tt,  —  y^x. 

3.  Find  the  angle  subtended  at  the  centre  of  a  circle  of  radius 
3  inches  by  an  arc  of  5  inches. 

4.  If  the  radius  of  the  earth  is  3,960  miles,  find  the  angle  at 
the  centre  between  radii  drawn  to  two  points  of  the  surface 
200  miles  apart. 

6.  Coordinates.  Let  X'OX  and  Y'OY  be  two  perpendicular 
lines  intersecting  at  0.  These  lines  divide  the  plane  into  four 
parts  called  quadrants.  The  portion  of  the  plane  above  X'OX 
and  to  the  right  of  Y'OY  is  called  the  first  quadrant.     The 

second  quadrant  includes 
the  portion  of  the  plane 
above  X'OX  and  to  the 
left  of  Y'OY,  the  third 
quadrant  that  below 
X'OX  and  to  the  left  of 
Y'OY,  and  finally  the 
fourth  quadrant  that  be- 
low X'OX  and  to  the 
right  of  Y'OY.  By  long 
usage  it  has  become  cus- 
tomary to  consider  the 
directions  OX  and  OY 
as  positive.  In  what  fol- 
lows we  shall  therefore  consider  horizontal  lines  as  positive 
when  drawn  to  the  right,  and  vertical  lines  as  positive  when 
drawn  upwards. 


Y 

N 

P 

7-- 

y 

^    "V 

X 

0 

X 

'M 

^  A 

V. 

Y' 

Fig.  4. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


Let  now  P  be  any  point  of  the  plane,  and  let  MP  and  NP 
be  the  perpendicular  distances  of  P  from  OX  and  OF  respectively. 
If  the  point  P  be  given,  the  distances  NP  and  MP  (or  the  equi- 
valent pair  OM  and  MP)  can  be  measured.  Moreover  OM  and 
MP  cannot  have  the  same  pair  of  values  for  two  different  points. 
This  is  obviously  true  if  the  two  points  lie  in  the  same  quadrant. 
If  on  the  other  hand  they  lie  in  different  quadrants,  one  or 
both  of  the  quantities  OM  and  MP  must  have  different  signs 
in  the  two  quadrants. 

For  convenience  we  shall  denote  the  horizontal  distance  OM 
by  X,  and  the  vertical  distance  MP  by  y.  The  two  numbers 
X  and  y  are  called  the  coordinates  of  the  point  P.  If  P  be  in 
the  first  or  fourth  quadrant  OM  is  measured  to  the  right. 
Hence  x  is  positive  if  P  be  in  the  first  or  fourth  quadrant.  If 
P  be  in  the  second  or  third  quadrant  OM  is  measured  to  the 
left,  and  hence  x  is  negative  if  P  be  in  either  of  these  quadrants. 
Similarly  y  is  positive  in  the  first  and  second  quadrants,  for  in 
these  quadrants  MP  is  measured  upwards.  In  the  third  and 
fourth  quadrants  MP  is  measured  downwards,  and  therefore 
in  these  quadrants  y  is  negative. 

Example.  On  a  figure  draw  the  points  for  which  x  =  2, 
y  =  S;  x=-2,  y  =  S;  x=-2,  y=-S;  x  =  2,  y=-3. 

To  construct  the  first  of 
these  points  we  must  make 
0M  =  x  =  2.  Then  M  is 
two  units  to  the  right  of  0 
on  the  line  OX.  Also  MP 
=  y  =  S.  We  must  then 
draw  MP  perpendicular  to  - 
OX  at  M  and  mark  off  three 
units  above  M  on  this  line. 

To  construct  the  second 
point  we  must  make  OM 
=  x=  —2.  Then  M  is  two 
units  to  the  left  of  0  on  the 
line  OX'.  The  required 
point  is  then  three  units  above  M.  The  other  two  points  may 
be   similarly  constructed.      They  will    be  found  to  lie  in  the 


X — 2 

2/=3 


x-^2 


x^"i 
2/  =  3 


!r=2 
y — 3 


Y 
Fig.  5. 


6 


ELEMENTS   OF   PLANE   TRIGONOMETRY. 


third  and   fourth   quadrants   respectively.      The   four   points 
evidently  form  the  vertices  of  a  rectangle  whose  centre  is  at  0. 

Example.     Construct  the  points  for  which 
(a)  x  =  l,   y  =  2',  (b)  x=-3,  y  =  5; 

(c)  x=-3,  y=-S;  (d)  x  =  6,  y=-5, 

7.  Trigonometric  Functions.  In  general  one  quantity  is 
said  to  be  a  function  of  another  quantity  if  the  value  of  the 
former  depends  only  upon  that  of  the  latter.  As  a  simple 
example  suppose  that  a  man  is  walking  at  the  rate  of  four  miles 
per  hour,  and  that  he  walks  for  t  hours.  If  x  denote  the  dis- 
tance he  walks,  we  evidently  have  the  relation  x  =  4^.  The  value 
of  X  depends  only  upon  that  of  t.  Hence  we  say  that  a;  is  a  func- 
tion of  t. 

Let  us  now  as  in  page  4  have  two  perpendicular  lines  X'OX 
and  YVY.  With  OX  as  initial  line  draw  the  angle  XOP. 
This  angle  is  said  to  be  in  the  first,  second,  third,  or  fourth 
quadrant  according  as  the  terminal  line  OP  falls  in  one  or  other 


Fig.  6. 


of  these  quadrants.  In  any  one  of  these  cases  take  a  point  P 
on  the  terminal  hne  of  the  angle,  and  let  fall  the  perpendicular 
MP  from  P  upon  OX.  Also  let  Pi  be  any  other  point  on  OP, 
and  let  the  corresponding  perpendicular  be  MiPi.  Any  one 
of  the  triangles  OMP,  OMiPi,  etc.,  is  called  a  reference  triangle 
for  the  angle  XOP. 

The  two  right  triangles  OMP  and  OMiPi  are  similar,  and 
hence  the  ratio  of  any  two  sides  of  the  triangle  OMP  is  equal 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  7 

to  the  ratio  of  the  corresponding  sides  of  the  triangle  OMiPi. 
Hence  the  values  of  these  ratios  depend  not  upon  the  position 
of  the  point  P  upon  OP,  but  only  upon  the  size  of  the  angle 
XOP.  They  are  then  functions  of  this  angle.  Let  the  angle 
XOP  be  denoted  by  6.  Also  let  r  denote  the  length  of  OP,  and 
X  and  y  the  values  of  OM  and  MP  respectively.  From  the 
three  quantities  x,  y,  r  we  can  form  ratios  all  of  which  we  have 
just  seen  are  functions  of  the  angle  6.  We  define  these  func- 
tions as  follows: 

,^    MP  '  n    y 

sme  01  6  =  ^^^      or     sm  9  =  -, 
OP  r ' 

r.       OM  .       X 

cosme  ot  0  =  y^^      or    cos  0  =  - , 
OP  r 

MP  V 

tangent  of  ^  =  ^^^7     or    tan  6  =  -, 

cotangent  oi  6  =  j^      or    cot  ^  =  -, 

^    fa        OP  .      r 

secant  of  6  =  :p:r^rf     or    sec  d  =  -, 
OM  X 

cosecant  oi  6  =  irrp     or    esc  6  =  ~, 

,     .  en        1        OM  .       ..        X 

versed  sme  oi  B  =  I  — t^^^    or    vers  6  =  i . 

OP  r 

8.  The  Signs  of  the  Trigonometric  Functions.  Since  the 
positive  direction  of  the  terminal  line  of  6  is  from  0  to  P,  the 
value  of  r  is  in  all  cases  positive.  The  signs  of  the  trigonometric 
functions  will  then  depend  only  upon  the  signs  of  x  and  y. 

The  Sine.  The  sine  will  be  positive  whenever  y  is  positive; 
that  is,  sin  ^  is  a  positive  number  if  6  be  an  angle  in  the  first  or 
second  quadrant.  In  the  third  and  fourth  quadrants  y  is  nega- 
tive and  therefore  sin  ^  is  a  negative  number  if  6  be  an  angle  in 
the  third  or  fourth  quadrant. 

The  Cosine.  The  cosine  of  6  will  be  positive  or  negative  ac- 
cording as  X  is  positive  or  negative.  Hence  cos  6  is  positive  in 
the  first  and  fourth  quadrants,  and  negative  in  the  second  and 
third. 

The  Tangent.     The  tangent  of  0  will  be  positive  when  and  only 


8 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


when  X  and  y  are  of  the  same  sign,  for  then  only  can  their  ratio 
be  positive.  Hence  tan  6  is  positive  in  the  first  and  third  quad- 
rants, and  negative  in  the  second  and  fourth. 

Let  the  student  similarly  consider  the  cases  of  the  cotangent 
secant,  and  cosecant.  We  may  arrange  the  results  of  this  sec- 
tion in  the  form  of  a  table  as  follows: 


sine,  cosecant, 
cosine,  secant, 
tangent,  cotangent. 


I 

II 

Ill 

+ 

+ 



+ 

— 

— 

+ 

— 

+ 

IV 


+ 


9.  Given  One  Function  of  an  Angle  to  Find  all  the  Other 
Functions  of  This  Angle.  Consider  the  following  problem. 
Given  that  d  is  an  angle  in  the  second  quadrant  and  that  cos  6 
=  —  f ,  find  all  the  other  functions  of  d. 

We  have 


cos  0  =  -  = 
r 


In  constructing  the  reference  triangle  for  6  we  may  take  for 
OP  any  convenient  length.  Hence  we  may  put  r  =  5,  and  so 
the  conditions  of  our  problem  give  a;  =  —  3.     We  shall  now 

try  to  construct  an  angle  9 
in  the  second  quadrant  such 
that  X  =  —  S  and  r  =  5.  On 
XVX  layoff  OM  =  x=-S. 
Then  M  is  the  point  three 
units  to  the  left  of  0.  At 
M  erect  a  perpendicular  to 
XVX.  With  0  as  centre 
draw  the  arc  of  a  circle  of 
radius  5  cutting  the  per- 
pendicular at  P.  The  angle 
6  =  XOP  satisfies  the  conditions  of  the  problem.  The  length 
of  MP  may  be  calculated  from  the  relation  MP^  =  OP''  -  OMK 
This  gives  MP^  =  25  -  9  =  16,  and  therefore  MP  =  ±  4. 
But  MP  is  drawn  in  the  positive  direction  and  therefore 
MP  =  4. 


r 

P< 

/ 

\tV 

^ 

^ 

\ 

y— 

-3             \ 

o 

Fig.  7. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 
Hence  we  have 
sin0 


r      5'                  2/4'                  X 

4 
3' 

cot  0  =  -  =  —  . ,     sec  ^  =  -  =  —  -. 
2/4'                  X          S 

In  an  exactly  similar  way  any  problem  of  this  type  may  be  solved. 
10.  Exercise  II. 

1.  Find  all  the  functions  of  6  when  given  that 

(a)  6  is  an  angle  in  the  third  quadrant  and  tan  ^  =  f . 

(6)  d  is  an  angle  in  the  second  quadrant  and  sec  ^  =  —  yf . 

(c)  0  is  an  angle  in  the  fourth  quadrant  and  sin  ^  =  —  |. 

{d)  cos  6  =  —  ^  and  sin  6  is  negative. 

(e)  sin  ^  =  1^  and  sec  6  is  negative 

(/)  vers  0=1  and  tan  6  is  positive. 

(g)  tan  0  =  —  3  and  cos  6  is  positive. 

(h)  CSC  6  =  -/-  and  sec  6  is  negative. 

(i)  sin  0  =  —  Y  and  vers  0  is  greater  than  1. 

(j)  vers  0=1  and  esc  0  is  negative. 

2.  If  0  is  an  angle  in  the  fourth  quadrant  and  tan  0  =  —  |, 
find  the  value  of  (sin  0  +  4  cos  d)  (vers  (9  -  2  tan  ey. 

3.  If  0  is  an  angle  in  the  second  quadrant  and  sin  0  =  f , 
find  the  value  of  i^tan  d  +  vers  6  (cos  6  —  2  sec  6). 

4.  State  which  of  the  following  problems  are  possible,  and 
in  those  cases  which  have  solutions  find  all  the  functions 
oie: 

(a)  cosO  =  —  f ,  tan  0  =  f , 

-l/5 
(6)  sec  0  =  1,  tan  d  =  — - — , 

(c)  sin  0  =  —  I,  CSC  0  =  2, 
(c?)    CSC  0  =  i, 

(e)  CSC  0  =  3,  cos  0  =  ^-. 

o 


5.  Determine  the  signs  of  the  following  quantities:  sin  150°, 
tan  310°,  CSC  50°,  sec  123°,  sec  -^,  cot  — . 

11.  Variation  in  Size  of  the  Trigonometric  Functions.    With 


10 


ELEMENTS   OF   PLANE   TRIGONOMETRY. 


the  vertex  of  the  angle  XOP  as  centre  draw  a  circle  of  unit 
radius.  Let  this  circle  cut  OX,  OP,  and  OY  in  A,  P  and  B 
respectively.  At  A  and  B  draw  tangents  to  the  circle  to  meet 
OP  produced  in  T  and  S  respectively. 


Fig.  8. 


We  have  from  the  similar  triangles  OMP,  OAT,  and  SBO 

MP^MP 
OP  ~    1 

MP     AT 


sin  B  = 


tarn  6 


sec  a  = 


=  MP, 
AT 


^      OM     OM       ^,, 
cos  (9  =  ^  =  -^  =  OM, 


OM     BS 


■DO 

OM~OA~    1    =^^'  ^^^^=^^  =  5i"T  "^^' 


OP^^OT^OT  _  _  OP  ^OS^OS 

OM     OA       1    ~^^'    ^^^^~  MP     OB       1 


=  0S. 


vers  d  =  1 -cos  6  =  1- OM  =  OA-OM  =  MA, 

The  numerical  value  of  the  ratio  sin  6  is  equal  to  the  number 
of  units  of  length  of  the  line  MP,  where  the  radius  of  the  circle 
is  taken  as  this  unit;  and  similarly  for  the  other  functions. 
Observe  that  the  signs  of  the  functions  of  any  angle  may  be 
read  off  from  this  figure.  Such  a  function  is  positive  or  negative 
according  as  the  line  corresponding  to  it  is  drawn  in  the  positive 
or  negative  direction.  Thus  if  B  be  an  angle  in  the  second 
quadrant,  AT  is  drawn  downwards.     Hence  the  tangent  of  an 


ELEMENTS   OF  PLANE  TRIGONOMETRY.  11 

angle  in  the  second  quadrant  is  negative.  These  results  will 
be  found  to  agree  with  those  of  page  8. 

Let  us  now  consider  the  changes  in  the  values  of  the  trigo- 
nometric functions  as  the  angle  B  increases  from  0°  to  360°. 

The  Sine.  As  B  approaches  0°  the  length  of  the  line  MP  ap- 
proaches zero.  We  have  therefore  sin  0°  =  0.  As  ^  increases 
from  0°  to  90°  MP  continually  increases  from  0  to  OB  or  1. 
Therefore  sin  90°  =  1,  and  the  sine  of  any  angle  in  the  first 
quadrant  is  a  "positive  number  less  than  unity.  Now  as  B  increases 
from  90°  to  180°  MP  decreases  from  1  to  0  remaining  positive. 
Hence  sin  180°  =  0,  and  the  sine  of  an  angle  in  the  second 
quadrant  is  a  positive  number  less  than  unity.  Similarly  as  B 
increases  from  180°  to  270°  MP  becomes  negative  and  decreases 
from  0  to  —  1.  Then  sin  270°  =  —  1,  and  the  sine  of  an  angle 
in  the  third  quadrant  is  a  negative  number  between  0  and  —  1. 
In  a  similar  manner,  as  B  increases  from  270°  to  360°  MP 
remains  negative  and  increases  in  value  from  —  1  to  0.  Hence 
sin  360°  =  0,  and  the  sine  of  an  angle  in  the  fourth  quadrant 
is  a  negative  number  between  —  1  and  0. 

The  Tangent.  When  B  =  0°  we  have  AT  =  0,  and  hence 
tan  0°  =  0.  AsB  increases  from  0°  to  90°  AT  increases  without 
limit.  For  as  B  approaches  90°  the  line  OT  approaches  the 
position  of  a  parallel  to  AT.  Hence  when  B  =  90°,  AT  = 
tan  90°  =  00 .  The  tangent  of  an  angle  in  the  first  quadrant  is 
therefore  a  positive  number,  and  may  have  any  value  between 
0  and  00 .  As  B  enters  the  second  quadrant  A  T  becomes  nega- 
tive, and  as  the  angle  approaches  180°  the  length  oi  AT  ap- 
proaches 0.  Hence  the  tangent  of  an  angle  in  the  second 
quadrant  is  a  negative  number,  and  may  have  any  value  between 
—  00  and  0.  In  the  third  quadrant  the  tangent  again  becomes 
positive,  and  increases  in  value  from  0  at  180°  to  oo  at  270°. 
Finally  as  B  increases  from  270°  to  360°  tan  B  becomes  negative 
and  increases  in  value  from  —  oo  at  270°  to  0  at  360°. 

12.  Exercise  III. 

1.  What  angle  between  0°  and  360°  has  the  greatest  sine? 
The  least  sine?     The  greatest  cosine? 

2.  As  B  increases  from  0°  to  360°  explain  how  the  following 
functions  vary:  (a)  The  cosine,  (b)  the  cotangent,  (c)  the  secant, 
(d)  the  cosecant,  (e)  the  versed  sine. 


12  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

3.  For  what  values  of  6  is  sin  6  =  cos  6?     sin  ^  =  —  cos  6? 

4.  Given  an  angle  in  the  first  quadrant,  show  on  a  figure 
how  to  draw  an  angle  in  the  second  quadrant  such  that  the 
sines  of  the  two  angles  shall  be  equal.  Could  the  same  problem 
be  given  with  cosines  instead  of  sines? 

5.  Given  an  angle  6  in  the  first  quadrant,  in  what  quadrant 
must  <f)  be  if  tan  6  =  tan  0?  On  a  figure  show  how  to  construct 
<f)  when  given  6. 

13.  Periodic  Functions.  A  body  is  said  to  have  a  periodic 
motion  if  it  repeats  the  same  cycle  of  positions  over  and  over 
after  equal  intervals  of  time.  Thus  the  motion  of  the  earth 
around  the  sun  is  a  periodic  motion  since  every  year  it  repeats 
the  same  relative  set  of  positions.  The  length  of  time  that 
elapses  before  the  same  position  is  again  reached  is  called  the 
period.  In  a  similar  manner  a  function  is  said  to  be  periodic 
if  its  value  for  a  given  value  of  the  variable  is  always  the  same 
as  its  value  when  the  variable  is  increased  by  a  certain  constant. 
The  amount  by  which  the  variable  must  be  increased  in  order  to 
have  the  function  repeat  its  original  value  is  known  as  the  period 
of  the  function. 

In  Art.  11  we  considered  the  changes  in  the  values  of  the 
trigonometric  functions  as  the  angle  increased  from  0°  to  360°. 
If  now  the  angle  increase  beyond  360°  the  terminal  line  OP 
will  evidently  run  through  the  same  set  of  positions  that  it 
occupied  when  the  angle  was  between  0°  and  360°.  The  refer- 
ence triangles  for  the  two  angles  6  and  360°  +  6  will  therefore 
be  exactly  the  same.  It  follows  that  all  the  trigonometric 
functions  of  360°  +  0  are  exactly  the  same  as  the  corresponding 
functions  of  d.  In  other  words  the  trigonometric  functions 
are  periodic  functions  of  period  360°.  As  a  matter  of  fact  it 
will  be  seen  later  that  the  tangent  and  cotangent  are  periodic 
functions  of  period  180°,  for  it  will  be  shown  that  tan  (180°  +  d) 
=  tan  e  and  that  cot  (180°  -\- 6)  =  cot  6.     (Page  36,  ex.  7.) 

14.  Acute  Angles ;  Right  Triangles.  Let  0  be  an  acute  angle, 
that  is  an  angle  in  the  first  quadrant.  The  six  trigonometric 
functions  of  6  are  positive.  The  angle  6  is  one  of  the  acute 
angles  of  the  triangle  OMP.  In  this  triangle  OP  or  r  is  the 
hypotenuse,  OM  or  x  is  the  side  adjacent  to  the  angle  6,  and  MP 


ELEMENTS   OF  PLANE  TRIGONOMETRY. 


13 


or  y  is  the  side  opposite  to  this  angle.     We  then  have  for  the 
six  trigonometric  functions  of  an  acute  angle: 

y 


Fig.  9. 


•    /»      V      opposite  side 

sm  0  =  -  =  ,f^    , , 

r       hypotenuse 

■      adjacent  side 


^  _  r  _   hypotenuse 
y      opposite  side' 


cos  6 


sec  9 


hypotenuse  ' 

tan  0  =  ^  =  oPPQs^^e  ^^^^ 
X      adjacent  side' 

In  problems  concern- 
ing right  triangles  these 
relations  are  very  useful. 
Thus  let  ABC  be  a  tri- 
angle right-angled  at  C. 
Also  let  the  lengths  of 
the  sides  be  denoted  by 
a,  b  and  c,  where  a  is 
the  side  opposite  the  angle  A,  etc.     We  have 

sin  A 


r  _    hypotenuse 
X      adjacent  side' 

,  ^     X      adjacent  side 

cot  e=  -  =  —^ — TT — 73-. 

y      opposite  side 


Fig.  10 


a 


opposite  side 
hypotenuse       c' 

and  similarly  for  the  other  functions  of  A. 

In  the  same  way  we  may  find  the  functions  of  the  angle  B. 
For  this  angle  of  course  b  is  the  opposite  side  and  a  the  adjacent 
side. 

Hence 


tan  B 


opposite  side  _h  „ 

adjacent  side      a' 


hypotenuse    _  c    , 
adjacent  side      a' 


14 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


Observe  that 
sin  A 


cos  B  =  -,     tan  A  =  cot  B  =  t,  etc. 


Hence  since  B  =  90°  —  A,  we  have  the  following  relations: 


sin  A  =  cos  (90°  -  A), 
tan  A  =  cot  (90°  -  A), 
sec  A  =  CSC  (90°  -  A), 


cos  A  =  sin  (90°  -  A), 
cot  A  =  tan  (90°  -  A), 
CSC  A  =  sec  (90°  -  A). 


Of  the  two  functions  sine  and  cosine  either  is  said  to  be  the 
co-function  of  the  other.  Similarly  the  tangent  and  cotangent, 
and  the  secant  and  cosecant  are  called  co-functions.  The  six  re- 
lations above  may  be  expressed  as  follows:  Any  trigonometric 
function  of  an  angle  A  is  equal  to  the  co-function  of  the  complement 
of  A.  In  this  section  we  have  proved  this  rule  only  when  A 
is  an  acute  angle.  The  rule  however  is  true  for  all  values  of 
A.     [Art.  31.] 

15.  Functions  of  45°.  Let 
ABC  be  a  triangle  right-angled 
at  C,  and  let  the  angle  A  be  equal 
to  45°.  Then  B  is  also  equal  to 
45°,  since  the  sum  of  the  acute 
angles  of  a  right  triangle  is  equal 
to  90°.  The  triangle  is  there- 
fore isosceles  and  a  =  h.  Hence 
c  =  T/a2  +  62  =  i/2a2  =  a\/2. 
We  therefore  have  the  follow- 
ing values  for  the  functions  of 
45°: 


sin  45°  =  -  =  -— - 
c      aV2 


1 

y'2 


W2, 


cos  45°  =  -  =  ■ — -=-  =  —^  =  ii/2, 
c      aV2      V2       ^    ' 


tan  45°  = 


1. 


16.  Functions  of  30°  and  60°.  Let  ABC  be  a  triangle 
right-angled  at  C,  and  let  the  angle  A  be  equal  to  30°  Produce 
the  line  BC  to  D  making  DC  =  CB,     The  triangle  ABD  is 


ELEMENTS   OF  PLANE   TRIGONOMETRY. 


15 


equilateral  since  each  of  its  angles  is  equal  to  60°.     Hence 


Also 


and  hence 


DB  =  AB  =  c,     and    CB  =  a=~. 


62  =  c2 


Fig.  12. 
We  therefore  have 

sin  A  =  sin  30°  =  -  =  i,    sin  B  =  sin  60°  =  ^  =  iv% 
c  c 

cos  A  =  cos  30°  =  -  =  ii%     cos  B=  cos  60°  =  "  =  |, 
c  c 

tan  A  =  tan  30°  =  ^  =  iv%    tan  5  =  tan  60°  =  -  =  v'S. 
0  a 

17.  Exercise  IV. 

1.  Find  the  values  of  the  cotangent,  •  secant,  cosecant,  and 
versed  sine  of  each  of  the  following  angles:  30°,  45°,  60°. 

2.  Prove  that   cos  60°  =  (cos  30°  +  sin  30°)    (cos   30°  - 
sin  30°). 


3.  Prove  that  sin  45°  cos  30°  -  cos  45°  sin  30' 


'-^f^ 


COS  30° 


4.  Find  the  values  of  sin  120°,  cos  210°,  tan  135°,  sin  300°, 
vers  330°.* 

*  Note  that  the  reference  triangles  for  these  angles  are,  except  for  alge- 
braic signs,  the  same  as  those  for  the  angles  30°,  45°,  and  60°. 


16 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


5.  Find  the  value  of 


cos  120°  (tan  225°  -  sin  330°) 


cos  300°  (sin  240°  +  cos  150°)* 

6.  Express  each  of  the  other  functions  of  6  in  terms  of  esc  6. 

Hint.     Construct  a  right  triangle  ABC  in  which  the  angle 

BAC  equals  d.     Then  esc  d  =  AC/BC.     Hence,  if  we  take  the 

length  of  BC  as  the  unit  of  measure,  esc  6  =  AC  and  AB  = 

Vac^  -bc^=  i/csc2  e  -  1.* 


The  lengths  of  the  sides  of  ABC  are  then  as  denoted  in  the 
accompanying  figure. 
Hence, 

BC 


sin  B 


tan^  = 


AC 
BC 


1 

CSC  e' 


COS0  = 


AB      i/csc20-l 


AC 


cscO 


1 


-,  etc. 


AB       i/csc2  0-l' 

7.  Express  each  of  the  other  functions  of  6  in  terms  of  (a) 
sin  d,  (6)  cos  e,  (c)  tan  0,  (d)  cot  0,  (e)  sec  0. 

18.  Polar  Coordinates.  Let  XV  X  be  a  given  straight  line 
and  0  a  given  point  on  it.  Then  the  position  of  any  point  P 
in  the  plane  of  the  paper  is  determined  when  the  angle  XOP 
and  the  distance  OP  are  given.     If  then  we  denote  the  angle 

*  The  symbol  csc^^  means  [esc  6]^  and  is  read  "cosecant  squared  of  ^." 
Similarly  sin^  6  means  the  square  of  the  sine  of  0,  etc. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


17 


XOP  by  d  and  the  distance  OP  by  r,  the  position  of  P  is  known 
when  the  values  of  6  and  r  are  given.  The  positive  direction 
of  OP  (=  r)  is  from  0  along  the  terminal  line  of  the  angle 
XOP  (=6).     Instead   of   de-  -^ 

fining  the  position  of   a  point  ^  ^ 

by  equations  such  as  ^  =  30°,    x- ^^ ^ ^ 

r  =  2,  it  is  customary  to  speak  Fig.  14. 

of  the  point  (2,  30°).  The  values  of  d  and  r  are  called  the 
polar  coordinates  of  the  point,  and  both  this  pair  of  values  and 
the  point  which  corresponds  to  them  are  denoted  by  the  symbol 
(r,  d).  Fig.  15  shows  the  points  (2,  30°)  and  (3,  240°).  To  con- 
struct the  point  (—3, 120°)  we  first  draw  the  terminal  line  of  the 


angle  120°  and  then  measure  a  distance  —  3  upon  this  line. 
But  the  positive  direction  of  this  line  extends  into  the  second 
quadrant  and  hence  a  negative  distance  must  be  measured  along 
the  production  of  this  terminal  line  into  the  fourth  quadrant. 
Thus  the  point  is  constructed  as  in  Fig.  15. 

Example.  Construct  the  following  points :  (2,  65°) ,  ( —  2,  25°) 
(3,  110°),  (-2,  225°),  (1,  240°),  (-3,  330°). 

19.  Numerical  Values  of  Sine,  Cosine,  Tangent,  Cotangent. 
In  the  following  table  will  be  found  the  values  correct  to  two 
2 


18 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


decimals  of  these  four  functions  for  every  five  degrees  from  0^ 
to  90°. 


o 

Sin. 

Tan. 

Cot. 

Cos. 

0 

0 

0 

OO 

1 

90 

5 

0.09 

0.09 

11.43 

0.996 

85 

10 

0.17 

0.18 

5.67 

0.98 

80 

1£ 

0.26 

0.27 

3.73 

0.97 

75 

20 

0.34 

0.36 

2.75 

0.94 

70 

25 

0.42 

0.47 

2.14 

0.91 

65 

30 

0.50 

0.58 

1.73 

0.87 

60 

35 

0.57 

0.70 

1.43 

0.82 

55 

40 

0.64 

0.84 

1.19 

0.77 

50 

45 

0.71 

1 

1 

0.71 

45 

Cos. 

Cot. 

Tan. 

Sin. 

o 

For  angles  less  than  45°  the  functions  are  found  at  the  top 
of  the  table  and  the  angles  in  the  left  hand  column.  For 
angles  greater  than  45°  the  functions  are  found  at  the  bottom 
of  the  table  and  the  angles  in  the  right  hand  column.  Thus  we 
find  from  the  table  sin  15°  =  0.26,  cos  70°  =  0.34,  etc. 

We  may  also  find  the  val- 
ues of  the  functions  of  angles 
greater  than  90°  in  the  fol- 
lowing manner.*  Let  us  be 
required  to  find  the  value 
of  cos  120°.  First  construct 
the  angle  XOP  =  120°  and 
the  angle  XOQ  =  POX'  = 
180° -XOP  =  60°.  If  then 
we  take  OP  =  OQ  and  con- 
struct the  reference  triangles 
OMP  and  ONQ,  it  is  evi- 
dent that  these  triangles  are 
equal  in  all  respects  except  that  ON  is  positive  while  OM  is 
negative,  so  that  we  have  OP  =  OQ,  MP  =  NQ,  OM  =  -  ON. 
Hence  cos  120°  =  OM/OP  =  -  ON/OQ  =  -  cos  60°  =  -  0.5. 
Similarly  sin  120°  =  MP/OP  =  NQ/OQ  =  sin  60°  =  0.87  and 
tan  120°  =  MP/OM  =  NQ/{-  ON)  =  -  NQ/ON  =  -  tan  60° 
=  -  1.73. 

*  For  a  more  complete  discussion  of  this  problem  see  section  31. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


19 


In  a  similar  manner  the  functions  of  angles  in  the  third  and 
fourth  quadrants  may  be  found.  Thus  the  reference  triangles 
for  the  angles  230°  and  50°  are  equal  except  for  the  algebraic 
signs  of  the  two  legs,  and  hence  the  functions  of  230°  may  be 
found  from  the  corresponding  functions  of  50°. 

20.  Graphs  of  Equations  in  Polar  Coordinates.  Let  us  be 
given  an  equation  involving  r  and  d,  for  example 

r  =  1  -  2  cos  0. 

If  we  assign  any  value  to  B,  say  20°,  the  equation  determines 


co° 


a  corresponding  value  for  r,  that  is,  1  —  2  cos  20°  =  1  — 
2(.94)  =  1  -  1.88  =  -  .88.  Thus  the  pair  of  values  r  =  -  .88 
6  =  20°  satisfy  the  given  equation.  To  this  pair  of  values  there 
corresponds  a  point  in   Fig.  17.     In  a  similar  manner  if  we 


20  ELEMENTS  OF  PLANE   TRIGONOMETRY. 

let  e  equal  70°,  we  find  r  =  1  -  2(0.34)  =  1  -  0.68  =  0.32 
and  to  this  pair  of  values  there  corresponds  another  point  in  the 
figure.  We  may  thus  construct  as  many  points  as  we  choose, 
the  coordinates  of  each  of  which  satisfy  the  given  equation. 
It  will  be  found  that  all  of  these  points  lie  upon  a  smooth  curve 
and  this  curve  is  called  the  graph  of  the  equation. 

To  find  the  graph  of  any  equation  it  is  well  to  first  construct 
a  table  as  follows.     Let  the  equation  be  r  =  1  —  2  cos  ^. 


e 

COS0 

2coa0 

r 

Point 

0° 

1 

2 

-1 

(-1,  0°) 

10° 

0.98 

1.96 

-0.96 

(-.96,  10°) 

20** 

0.94 

1.88 

-0.88 

(-.88,  20°) 

30° 

0.87 

1.74 

-0.74 

(-.74,  30°) 

40° 

0.77 

1.54 

-0.54 

(-.54,40°) 

50° 

0.64 

1.28 

-0.28 

(-.28,50°) 

60° 

0.5 

1 

0 

(0,  60°) 

70° 

0.34 

0.68 

0.32 

(0.32,  70°) 

80° 

0.17 

0.34 

0.66 

(0.66,  80°) 

90° 

0 

0 

1 

(1,  90°) 

100° 

-.17 

-.34 

1.34 

(1.34,  100°) 

110° 

-.34 

-.68 

1.68 

(1.68, 110°) 

180°  -1  -2  3  (3,180°) 

This  table  may  be  completed  by  the  student  for  angles  in 
the  third  and  fourth  quadrants.     Each  point  should  then  be 
located,  and   finally  a  smooth  curve  drawn  carefully  through 
successive  points. 
21.  Exercise  V. 
Draw  the  graphs  of  the  following  equations: 


1.  r  =  a  sin  ^ 

8.  r  =  2  sin  30 

15. 

r  =  sec  0 

2.  r  =  a  sin  2<? 

9.  r=l-sin2(9 

16. 

r2  =  sin  Q 

3.  r=a(l+sin(9) 

10.  r  =  1+2  cos  0 

17. 

r2=l+sin0 

4.  r  =  a(l  — sin  B) 

11.  r  =  sin  0+cos  6 

18. 

r2=l-2sin0 

5.  r'  =  2-cos  e 

12.  r  =  tan(9 

19. 

r2  =  2-sin0 

6.  r  =  2sin2  0 

13.  r=l-tan(9 

20. 

r2  =  sin2(9 

7.  r  =  l-2  COS0 

14.  r  =  sm  - 
2i 

ELEMENTS  OF  PLANE  TRIGONOMETRY. 


21 


22.  Graph  of  y  =  sin  x.  li  x  =  0°,  2/  =  sin  0°=  0;  similarly 
iix  =  30°,  y  =  sin  30°  =  -^.  In  this  way  we  may  obtain  as  many 
pairs  of  corresponding  values  of  x  and  y  as  we  wish.  Take  any 
convenient  length  on  the  X  axis  to  represent  30°.  Twice  that 
length  will  be  60°  and  similarly  any  other  angle  may  be  repre- 
sented by  a  length  on  the  X  axis.  Plot  the  points  (see  Art.  6) 
(0°,0),  (30°,  .5),  (45°,  .71),  (60°,  .87),  (90°,  1),  (120°,  .87),  (135°, 
.71),  (150°,  .5),   (180°,  0),  (210°,  -.5)  and  so  on  through  the 


third  and  fourth  quadrants.  Draw  a  smooth  curve  through  these 
points  and  we  shall  have  the  graph  oi  y  =  sin  x.  Evidently  as 
X  increases  beyond  360°  the  values  of  y  repeat  themselves  and 
the  complete  curve  is  an  endless  series  of  waves  exactly  like  the 
one  in  the  figure. 

23.  Doing  the  same  thing  with  y  =  cos  x  we  get  its  graph 
as  in  Fig.  18. 

24.  In  graphing  y  =  tan  x,  we  get  the  points  (0°,  0),  (30°,  .58), 
(45°,  1),  (60°,  1.73),  (90°,  00).  The  point  (90°,  cx))  raises  a 
new  difficulty,  to  get  over  which  we  must  remember  that  when 
X  is  between  60°  and  90°,  tan  x  is  positive.  Hence  the  curve 
goes  up  as  in  Fig.  20,  continually  getting  closer  to  the  line  AB 
but  never  actually  touching  it.  If  x  is  between  90°  and  120°, 
tan  X  is  negative;  so  the  curve  is  below  the  X  axis,  passes  up- 
ward through  (120°,  -  1.73),  (135°,  -  1),  (150°,  -  .58),  (180°, 
0),  (210°,  .58),  (240°,  1.73),  (270°,  00).  When  x  is  between 
240°  and  270°,  tan  x  is  positive,  so  that  the  curve  in  this  region 
is  above  the  X  axis.  When  a:  is  a  httle  larger  than  270°,  tan  x 
is  negative;  so  the  curve  appears  below  the  X  axis  and  passes 


22 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


upward  through  (300°,  -  1.73)    (330°,  -  .58),  (360°,  0),  etc. 
It  will  be  noticed  that  the  curve  from  180°  to  360°  is  exactly 
the  same  as  it  was  between  0°  and  180°.     (See  ex.  7,  p.  36.) 


Fig.  20. 

It  is  left  as  an  exercise  to  the  student  to  draw  the  part  of 
these  graphs  for  which  x  is  negative. 


CHAPTER  II. 

25.  Solution  of  Right  Triangles.  We  have  seen  in  articles 
15  and  16  that  it  is  possible  to  compute  the  values  of  the 
trigonometric  functions  of  30°,  45°  and  60°.  By  methods  in- 
volving more  knowledge  than  the  student  of  elementary  trigo- 


a=2i 


Fig.  21. 

nometry  possesses  it  is  possible  to  compute  the  values  of  the 
trigonometric  functions  of  any  desired  angle.  These  values 
are  collected  in  tables  so  arranged  that  any  function  of  a  given 
angle  may  be  readily  found.  These  are  called  tables  of  the 
Natural  Functions.  However  in  the  great  majority  of  actual 
problems  it  is  found  that  the  logarithms  of  these  functions  are 
more  easily  used  than  the  functions  themselves.  Tables  have 
therefore  been  computed  of  the  logarithms  of  each  function 
of  angles  from  0°  to  90°.  By  the  symbol  log  sin  39°  is  meant 
the  logarithm  of  that  number  which  is  found  to  be  the  value  of 
sin  39° ;  and  similarly  for  the  logarithms  of  the  other  functions. 

Let  us  now  have  a  triangle  ABC  right-angled  at  C.  In  this 
triangle  let  us  be  given  that  the  angle  A  is  equal  to  26°  34', 
and  that  the  length  of  the  side  a  is  24  ft.  It  is  required  to 
find  the  values  of  the  angle  B  and  the  sides  h  and  c. 

We   have   B  =  90°  -  A  =  90°  -  26°   34'  =  63°   26'.    And 

sin  A  =  sin  26°  34'  =  -  =  -, 
c       c 

tan  A  =  tan  26°  34' =  ^  =  ^. 

0  0 

23 


24  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

Solving  these  equations  for  the  required  parts  we  have 

24  ,      ,  24 

c  =  - — nno  nAf     and     0  = 


sin  26°  34'     ""^      "       tan  26°  34'* 

Referring  to  the  tables  of  natural  functions  we  find  that 
sin  26°  34'  =  0.44724  and  tan  26°  34'  =  0.50004.  Hence  upon 
performing  the  indicated  divisions  we  have 

c  =  53.662  ft.     and     b  =  47.996  ft. 
This  triangle  may  also  be  solved  by  logarithms. 

Thus  we  find  from  the  logarithmic  tables 

log  24  =  11.38021  -  10  log  24  =  1.38021 

log  sin  26°  34'  =    9.65054  -  10      log  tan  26°  34'  =  9.69900 
therefore  log  c  =     1.72967  and  log  h  =  1.68121 

and  hence      c  =  53.662  ft.         and  b  =  47.996  ft. 

This  problem  illustrates  the  method  of  solving  a  right  tri- 
angle. From  our  study  of  geometry  we  know  that  if  any  two 
parts  of  one  right  triangle  (one  of  which  at  least  is  a  side) 
be  equal  to  the  two  corresponding  parts  of  another  right  tri- 
angle, the  two  triangles  are  in  all  respects  equal.  In  other 
words  if  two  such  parts  be  given  the  remaining  three  parts  are 
determined.  One  of  the  purposes  of  the  study  of  trigonometry 
is  to  enable  the  student  to  compute  the  values  of  these  three 
parts.     Two  types  of  problem  only  need  be  considered. 

1.  Given  an  angle  and  a  side,  for  example  A,  a 

The  angle  B  may  be  found  by  the  relation  B  =  90°  —  A. 

To  find  b  make  use  of  a  function  of  the  given  atigle  which 
involves  the  given  side  a  and  the  required  side  b,  as  tan  A  =  a/b. 
Then  b  =  a/tan  A.  Hence  by  aid  of  the  logarithmic  tables 
we  may  find  the  value  of  b. 

To  find  the  side  c  make  use  of  a  function  of  the  given  angle 
A  which  involves  both  the  given  side  a  and  the  required  side  c, 
as  sin  A  =  a/c.  Hence  c  =  a/sin  A  and  we  may  find  the  value 
of  c. 

In  finding  both  b  and  c  make  use  of  the  two  given  parts  A, 
a.  It  would,  of  course,  be  possible  to  find  c,  after  having 
found  the  value  of  b  as  above,  by  means  of  the  relation  sin  B 
=  b/c.    This  method  has,  however,  the  serious  disadvantage 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  25 

that  any  error  we  may  have  made  in  finding  b  will  also  appear 
in  c. 

2.  Given  two  sides,  for  example  a,  b. 

The  ratio  of  the  given  sides  will  be  some  function  of  one  of 
the  acute  angles;  in  this  case  A  may  be  found  from  a/b  =  tan  A. 
Having  made  sure  that  A  is  correct,  the  question  is  finished 
as  in  the  first  case. 

26.  Angles  of  Elevation  and  Depression.  Let  us  have  two 
points  A  and  C  in  two  different  horizontal  planes.  For  example 
let  C  be  a  point  higher  than  A  as  in  the  figure.  Draw  the 
vertical  line  CC  through  C  and  let  this  line  cut  the  horizontal 
plane  through  A  in  C\  so  that  C  is  the  point  vertically  below 
C  and  at  the  same  level  as  A.  The  angle  C'AC  is  called  the 
angle  of  elevation  of  C  from  A,  Similarly  if  A  A'  is  the  vertical 
line  through  A,  and  A'  the  point  in  which  it  cuts  the  horizontal 
plane  through  C,  the  angle  A'CA  is  called  the  angle  of  depression 


Fig.  22. 

of  A  from  C.  It  is  obvious  that  the  angle  of  elevation  of  C 
from  A  is  equal  to  the  angle  of  depression  of  A  from  C.  Let 
B  be  a  third  point  and  let  the  vertical  line  BB'  meet  the  hori- 
zontal plane  through  A  in  5'.  The  plane  AB'C  is  horizontal. 
The  angle  B'AC^  is  defined  as  the  horizontal  angle  at  A  subtended 
by  B  and  C,  or  the  horizontal  angle  at  A  of  5  and  C. 

By  means  of  an  instrument  known  as  a  transit  surveyors  are 


26  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

enabled  to  measure  horizontal  angles  and  angles  of  elevation 
and  depression. 

By  the  horizontal  distance  between  two  points  is  meant  the 
distance  between  the  vertical  lines  through  the  two  points. 
Thus  in  Fig.  22  the  horizontal  distance  between  B  and  C  is 
B'C\  that  between  A  and  C  is  AC,  etc. 

27.  Exercise  VI. 

1.  A  flag-pole  stands  in  a  level  field.  The  angle  of  elevation 
of  the  top  of  the  pole  from  a  point  200.64  feet  from  the  base  is 
found  to  be  19°  28'  35''.     Find  the  height  of  the  pole. 

Let  C  be  the  base  and  B  the  top  of  the  pole,  and  A  the  given 
point  in  the  field.     In  the  right  triangle  ABC, 

A  =  19°  28'  35"    and    CA  =  h  =  200.64  ft. 


Fig.  23. 

Our  problem  then  consists  in  finding  the  value  of  CB  =  a 
when  given  A  and  h.     We  have 

tan  A  =  T    and  therefore    a  =  b  tan  A. 
b 

log  h  =  2.30242 

log  tan  A  =  9.54858  -  10 

log  a  =  1.85100 

Hence  a  =  the  height  of  the  flag-pole  =  70.958  ft. 

2.  From  a  point  A  on  one  bank  of  a  river  the  angle  of  eleva- 
tion of  the  top  of  a  tree  growing  on  the  opposite  bank  is  5°  53' 
20".  A  horizontal  line  AB  equal  to  206.45  ft.  is  measured 
along  the  river  bank  at  right  angles  to  the  line  from  A  to  the 
foot  of  the  tree.  The  horizontal  angle  subtended  at  B  by  A 
and  the  tree  is  found  to  be  72°  48'  5".  Find  the  height  of  the 
tree  and  the  distance  from  B  to  the  foot  of  the  tree. 

Let  C  and  D  denote  the  foot  and  top  of  the  tree  respectively. 
We  then  have  the  two  right  triangles,  ABC  right-angled  at  A, 


ELEMENTS  OF   PLANE  TRIGONOMETRY. 


27 


and  ACD  right-angled  at  C.  In  the  former  triangle  ABCj  we 
know  AB  =  206.45,  and  the  angle  ABC  =  72°  48'  5''.  We 
can  therefore  find  the  required  distance  BC.  From  this  triangle 
we  must  also  find  the  value  of  the  side  AC.    Having  found  AC 


Fig.  24. 

we  shall  have,  in  the  triangle  ACD,  the  side  AC  and  the  angle 
CAD  =  5°  53'  20". 


In  the  triangle  ABC 
AB 


cos  ABC  = 


Also 


tan  ABC  = 


BC 
AC 


and  therefore    BC  = 


AB 


cos  ABC 


AB 

log  AB  =  2.31482 
log  cos  ABC  =  9.47083 


and  therefore    AC  =  AB  tan  ABC. 


log  AB  =  2.31482 
log  tan  ABC  =  0.50931 


log  BC  =  2.84399  log  AC  =  2.82413* 

Hence  BC  =  distance  from  B  to  the  foot  of  the  tree  =  698.22 
feet. 

In  the  triangle  ACD 


tan  CAD  = 


CD 
AC 


and  therefore    CD  =  AC  tan  CAD. 


log  AC  =  2.82413  (as  found  above) 
log  tan  CAD  =  9.01344 


log  CD  =  1.83757 

Hence  CD  =  height  of  the  tree  =  68.797  feet. 

*  Note  that  it  is  not  necessary  to  find  the  value  of  AC,  since  this  dis- 
tance is  not  required,  and  the  loearithm  oi  AC  only  is  needed  in  the  rest 
of  the  work. 


28  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

3.  A  tower  stands  on  the  shore  of  a  river  206.8  feet  wide. 
The  angle  of  elevation  of  the  top  of  the  tower  from  the  point 
on  the  other  shore  exactly  opposite  to  the  tower  is  46°  31'  2h" , 
Find  the  height  of  the  tower. 

4.  The  angle  of  elevation  of  the  top  of  a  mountain  from  a 
point  of  observation  is  found  to  be  41°  54'  35''.  Find  the 
height  of  the  mountain  if  the  horizontal  distance  from  the 
given  point  to  the  top  is  known  to  be  5436.7  feet. 

5.  At  141.78  feet  from  the  base  of  a  tower,  and  on  a  level 
with  the  base,  the  angle  of  elevation  of  the  top  was  found  to 
be  54°  22'  55".     Find  the  height  of  the  tower. 

6.  A  rope  is  stretched  from  the  top  of  a  building  to  the 
ground.  The  rope  is  found  to  make  an  angle  of  58°  56'  10" 
with  the  horizontal,  and  the  building  is  61.004  feet  high.  Find 
the  length  of  the  rope. 

7.  From  one  end  of  a  bridge  the  angle  of  depression  of  an 
object  200.87  feet  down  stream  from  the  bridge  and  at  the 
water  line  is  found  to  be  21°  43'  55".  From  the  same  point 
the  angle  of  depression  of  an  object  at  the  water  line  exactly 
under  the  opposite  end  of  the  bridge  is  found  to  be  17°  22'  25". 
Find  the  length  of  the  bridge  and  its  height  above  the  river. 

8.  From  a  ship's  masthead  157.87  feet  high  the  angle  of 
depression  of  a  boat  is  27°  56'  25".  Find  the  distance  from 
the  boat  to  the  ship. 

9.  Two  flag-poles  are  known  to  be  61.502  and  43.124  feet 
high  respectively.  A  person  moves  about  until  he  finds  a 
position  from  which  the  top  of  the  nearer  pole  just  hides  that 
of  the  farther.  From  this  point  the  angle  of  elevation  of  the 
top  of  the  nearer  one  is  found  to  be  36°  21'  10".  Find  the 
distance  between  the  poles,  and  the  distance  from  the  observer 
to  the  nearer  pole. 

10.  From  the  foot  of  a  post  32.548  feet  high  the  angle  of 
elevation  of  the  top  of  a  steeple  is  63°  43'  50",  and  from  the 
top  of  the  post  the  angle  of  depression  of  the  base  of  the  steeple 
is  49°  12'  35".  Find  the  height  of  the  steeple  and  its  distance 
from  the  post. 

11.  The  angles  of  depression  of  the  top  and  foot  of  a  build- 
ing  seen   from   the    top   of    a   tower   98.253   feet   high   are 


ELEMENTS  OF  PLANE   TRIGONOMETRY.  29 

32°  21'  55"  and  59°  56'  10"  respectively.     Find  the  height  of 
the  building. 

12.  From  the  top  of  a  cliff  156.43  ft.  high  the  angles  of  depres- 
sion of  two  boats  at  sea,  each  due  north  of  the  observer,  are 
31°  52'  55"  and  17°  23'  35".     How  far  are  the  boats  apart? 

In  the  following  problems  ABC  represents  a  triangle  right- 
angled  at  C,  and  the  letters  a,  6,  and  c  represent  the  sides  of 
the  triangle  opposite  to  the  angles  A,  B,  and  C  respectively. 
In  each  problem  find  all  of  the  parts  which  are  not  given. 

13.  a  =  235.61,  h  =  687.99. 

14.  b  =  78352,  A  =  54°  55'  25". 

15.  c  =  2138.7,  A  =  29°  43'  45". 

16.  c  =  1.3276,  a  =  0.65998. 

17.  c  =  78.657,  B  =  81°  21'  25". 

18.  a  =  0.02659,  A  =  19°  54'  35". 

19.  b  =  568.02,  c  =  603.87. 

20.  a  =  528090,  A  =  63°  27'  55". 


CHAPTER  III. 
TRIGONOMETRIC  ANALYSIS. 

28.  Relations  among  the  Trigonometric  Functions.     Let  6 

be  an  angle  in  any  quadrant,  and  draw  the  reference  triangle 
OMP.    By  definition  we  have 


'^JT 


■^x 


Fig.  25. 


Hence 


sin  0  =  -     and    esc  B 
r 


CSC  B 


1 


sin0' 

Ir  the  same  way  we  may  show  that 

1 


cot0  = 


and 


sec  9  = 


tan^ 

1 
cos  d' 


(1) 

(2) 
(3) 


The  sine  and  cosecant  are  said  to  be  reciprocal  functions,  for 
either  one  of  them  is  the  reciprocal  of  the  other.  Similarly 
the  tangent  and  cotangent,  and  the  secant  and  cosine  are 
reciprocal  functions. 


We  also  have 


sin  0 
cos^ 


y 

X 

r 
30 


ELEMENTS  OF  PLANE   TRIGONOMETRY.  31 

and  hence 

^-'  =  tane.  (4) 

COS  e  ^ ' 

Similarly  [or  by  aid  of  equations  (2)  and  (4)]  we  have 
cos  d 


sin  d 


=  cot  e.  (5) 


Also  in  the  triangle  OMP  we  have  x^  -{-  y^  =  r^.  If  we 
divide  this  equation  by  r^  we  obtain 

(')■+(?)'■■■ 

Hence, 

cos2^  +  sin2  0  =  1.  (6) 

If  we  divide  the  equation  x^  -\-  y'^  =  r^  first  by  x^  and  then 
by  y"^  we  obtain  the  two  following  relations 

1  +  tan2  ^  =  sec2  d,  (7) 

cot2  0  +  1  =  csc2  d.  (8) 

29.  Identical  Equations.  If  A  denote  the  same  expression 
as  B,  or  one  which  can  be  transformed  into  B,  we  say  that  A 
is  identically  equal  to  B. 

The  notation  A  =  B  means  A  is  identically  equal  to  B. 
We  call  A  =  5  an  identical  equation  or  an  identity 

Thus  (x  +  2)2  -  4  =  x{x  +  4). 

For  (a;  +  2)2  -  4  =  ^2  +  4a;  +  4  -  4, 

=  a;2  +  4a:, 
=  x{x  +  4). 

It  follows  that  the  given  equality  is  an  identity  since  the 
first  expression  (x  +  2)2  —  4  can  be  transformed  into  the 
second  expression  x{x  +  4). 

The  eight  relations  of  the  previous  section  are  simple  examples 
of  trigonometric  identities.  For  in  each  of  these  relations  we 
have  seen  that  the  two  members  stand  for  the  same  expression. 

Consider  the  following  problem: 

Prove  that 

tan2  e  ^  -\-^  -  1. 
cos2  d 


32  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

We  have 

tan2  e  =  sec2  (9-1  (7),  p.  31 

^  ^^s^  ~  ^-  ^^^ 

The  given  relation  is  therefore   proved  true  since  the  first 
expression  can  be  transformed  into  the  second  expression. 
This  problem  may  also  be  proved  as  follows: 
-i—-  _  1  =  1  -  cos^  e 

cos2  e  cos2  e 

=  ^—  (e^ 

~cos2  0  ^^^ 

=  tan2  e.  (4) 

Hence  the  identity  is  a  true  one  since  the  second  expression  may 
be  transformed  into  the  first  expression. 

It  is  not  necessary  to  transform  one  of  the  given  expressions 
directly  into  the  other.  We  may  prove  the  identity  given  above 
in  the  following  way. 

Adding  1  to  each  member  of  the  assumed  identity,  we  have 

1 
cos^  d' 

This  is  true  since  each  member  is  identically  equal  to  sec^  6, 
the  first  by  (7)  and  the  second  by  (3),  section  28. 

It  must  be  observed  that  in  this  last  method  all  we  have  really 
proved  is  that  if  tan^  0  =  l/cos^  ^  -  1,  then  sec^  d  =  sec^  d. 
What  we  really  wish  to  prove  is  that  if  sec^  6  =  sec^  d  (which 
is  obvious),  then  the  given  identity  is  a  true  one.  That  this 
is  true  follows  from  the  fact  that  each  step  in  our  work  may  be 
reversed.  In  other  words  we  may  go  backwards  from  the  known 
relation  sec^  d  =  sec^  6  to  the  desired  relation 

tan2  6  ^  -^-  -  1. 
cos^  6 

Thus  sec2  e  =   sec2  d  gives  by  aid  of  (7)  and  (3) 

tan2  e  +  l^  — V^, 
cos^  B 


tan^  e  +  l  =  -^2 


and  hence,  upon  subtracting  1  from  each  member, 
as  required. 


tan2  0  ^  — V^  -  1, 
cos^  d 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


33 


30.  Exercise  VII.     Prove  the  following  identities: 

(1)  cos  d  tan  6  =  sin  6. 

(2)  cos  6  CSC  6  =  cot  6. 

(3)  tan  0  +  cot  0  =  sec  6  esc  6. 

(4)  cos2  ^(1  +  tan2  6)  =  1. 

(5)  cos  6  +  tan  6  sin  d  =  sec  6. 

(6)  sin  d  +  cos  6  =  sin  6  cos  0(sec  0  +  esc  d). 

(7)  (1  +  tan  d){l  +  cot  ^)  sin  6  cos  d  =  (sin  0  +  cos  Oy. 

(8)  (cot  0  sin  ey  +  sin2  (9  =  (sin  d  +  cos  sy  -  2  sin  ^  cos  ^. 

(9)  sec2  d  -  1  =  sin2  ^  sec^  d. 

(10)  csc2  0  -  1  =  cos2  d  csc2  0. 

(11)  sec  X  —  1  =  sec  x  (1  —  cos  x). 

(12)  cos  X  CSC  X  tan  x  =  sin  x  sec  x  cot  x. 

(13)  (tan  d  -  sin  oy  +  (1  -  cos  0)^  =  (sec  6  -  \y. 

(14)  (cot  e  +  esc  ^)^  -  ^-^^^^. 

(15)  sec^  ^  -  tan2  6  =  sec^  6  +  tan^  ^. 

(16)  (cos2  e  -  l)(cot2  0  +  1)  +  1  =  0. 

(17)  sin  x(l  +  tan  x)  +  cos  x(l  +  cot  x)  =  sec  x  +  esc  x. 

(18)  (sin  60°  -  sin  45°)  (cos  30°  +  cos  45°)  -  sin^  30°. 

(19)  (cos  30°  -  sin  30°)  (sin  60°  +  cos  60°)  =  sin^  45°. 

(20)  If  the  lengths  of  the  sides  of  a  right  triangle  be  in 
arithmetical  progression,  prove  that  the  sines  of  its  acute  angles 
are  J  and  f  respectively. 

(21)  One  leg  of  a  right  triangle  is  equal  to  half  of  the  hypote- 
nuse; prove  that  the  acute  angles  are  30°  and  60°  respectively. 

31.  To  Express  the  Functions  of  90°  ±  6,  180°  ±  6,  etc., 
in  Terms  of  B. 

90°  -  6,  Let  e  be  any  angle  such  as  XOP)  then  if  P'OY 
=  e,  XOP'  will  be  90° 
—  B.  Draw  the  trian- 
gles of  reference.  By 
construction  P'OY  =  B; 
therefore  OP'M'  =  MOP, 
Hence  the  right  triangles 
POM  and  OP'M'  are 
equal  in  all  respects;  in 
particular  OM'  =  MP  and  M'P' 
3 


>x 


34 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


M'P'      OM 


sin  (90°  -e)  =  ~^pr  =  op  ==  ^^^  ^> 

mc^o       M      OM'       MP        .     ^ 
cos  (90°  -6)  =  ^pT  =  ^p  =  sin  6, 


tan  (90   -^)  =  om^  =  Sp 


cot0. 


Find  cot  (90°  -  6),  sec  (90°  -  d),  and  esc  (90°  -  d). 

180°-^.  Let  ^  be  the  angle 
XOP)  then  if  P'OM'=  6,  XOP' 
will  be  180°  -  6.  Draw  the  tri- 
angles of  reference.  The  right 
triangles  0PM  and  OP'M'  are 
equal  in  all  respects.  That  is, 
-^  M'P'  =  MP  and  OM'  =  -  OM. 
Then 


•     none       M       ^'^'      ^^ 
sm  (180°  -e)  =  ^pT  =  -^ 


cos  (180°  -  e)  = 
tan  (180°  -  0)  = 


OM^ 
OP'  '' 

WP'^ 
OM' 


OM 
OP 

MP 
OM 


sin  Oj 
=  —  cos  0, 
—  tan  B. 


Find  cot  (180°  -  B),  sec  (180°  -  B)  and  esc  (180°  -  B). 

—  B.     Again,  in  Fig.  28,  the  triangles  of  reference  are  equal 
in  all  respects  and  we  have 
MP'  =  -  MP.    Then 


sin  (-  B)  = 
cos  (-  B)  = 
tan  (-  B)  = 


MP' 
OP' 

= 

MP 
OP 

= 

—  sin  Bf 

OM 
OP' 

= 

OM 
OP 

= 

cos  B, 

MP' 
OM 

= 

MP 

OM 

= 

-  tan  B. 

Fig.  28. 


ELEMENTS  OF   PLANE  TRIGONOMETRY. 


35 


The  solution  of  any  question  of  this  type  depends  on  our 
abiUty  to  draw  the  angles  and  their  triangles  of  reference  and 
then  to  properly  pair  the  sides 
of  these  triangles,  attention  be- 
ing paid  to  the  signs.  Since 
logarithmic  tables  give  only 
acute  angles  it  is  necessary  to 
use  one  of  these  relations  in 
order  to  find  the  logarithm  of  a 
function  of  an  angle  greater 
than  90°.  The  most  useful  ones 
are  those  for  180°  -  6.  In  the 
examples  worked  out  6  was 
taken  as  an  acute  angle;  the 
same  results  would  be  obtained 

no  matter  what  the  size  of  B,  the  method  of  proof  being  the 
same  in  every  case. 

As  an  example,  let  us  take  90°  -\-  B^  6  being  an  obtuse  angle 
such  as  XOP.  Draw  POP'  =  90°;  then  XOP'  =  90°  +  B. 
Since  POP'  =  90°  and  MOP'  +  OP'M'  =  90°,  by  taking  away 
the  common  part  M'OP'  we  have  that  POM  =  OP'M',  There- 
fore the  triangles  OPM  and  OP'M'  are  equal  in  all  respects; 
in  particular  OM'  =  -  MP  and  M'P'  =  OM.     Then 


Fig.  29. 


sin  (90°  +  B)  = 


M'P^ 
OP' 


OM 
OP 


cos  B, 


cos  (90°  +  (9)  = 


0M[ 
OP' 


MP  .    ^    , 

-^  =  -  sm  B,  etc. 


The  results  of  all  these  are  included  in  the  following  rule: 
Any  Junction  of  90°  ±  B  {or  270°  ±  B)  is  equal  to  its  co-function 
of  B,  and  any  function  of  —  B,  180°  ±  0  or  360°  ±  B  is  equal  to 
the  same  function  of  B,  the  proper  sign  being  used  in  each  case.  To 
fix  the  sign,  suppose  B  to  he  acute  and  use  the  sign  of  the  function 
in  question  for  the  quadrant  in  which  the  compound  angle  falls. 

Thus  to  get  tan  (270°  +  (9),  we  say  that  with  B  acute  270° 
+  B  would  be  in  the  fourth  quadrant,  in  which  the  tangent  is 
negative.     Hence  tan  (270°  +  0)  =  -  cot  B. 


36 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


32.  Exercise  VIII. 

1.  By  a  figure,  prove  that  sin  (90°  +  ^)  =  cos  6  and  that 
cos  (90°  +  e)  =  -  sin  d.     Find  tan  (90°  +  6). 

2.  Find  sin  (180°  +  6),  cos  (180°  +  d),  and  tan  (180°  +(9) 
in  terms  of  d. 

3.  Find  tan  160°  10'  in  terms  of  19°  50'. 

4.  Find  log  sin  132°  20',  log  tan  200°  15',  log  cos  312°  18' 
and  log  sin  150°. 

5.  Prove  that  sin  160°  =  cos  70°  and  that  cos  110°  =  sin 
200°. 

6.  Given  sin  140°  =  a/b  find  sin  40°,  cos  130°  and  cos 
310°. 

7.  Prove  that  the  tangent  and  cotangent  are  periodic  func- 
tions of  period  180°  [p.  12]. 

33.  Inverse  Functions.  If  sin  ^  =  a  and  it  is  desired  to  draw 
attention  to  6,  one  would  use  the  phrase  "6  is  an  angle  whose 
sine  is  a."  This  is  written  d  =  arcsin  a.*  Similarly  6  =  arctan  h 
means  that  6  is  an  angle  whose  tangent  is  h.  Such  functions  are 
called  inverse  trigonometric  functions;  they  are  also  called 
circular  functions.  For  a  given  value  of  d  only  one  value  of  a 
is  possible;  but  on  the  other  hand  for  a  given  value  of  a,  there 
are  infinitely  many  values  of  6.  Thus  if  ^  =  arcsin  ^,  6  is  30° 
or  150°  or  any  angle  obtained  from  these  by  adding  multiples 
of  360°.  Similarly  arctan  t/3  is  60°  or  any  angle  obtained 
from  60°  by  adding  multiples  of  180°.  Of  all  these  values,  the 
smallest  positive  one  is  called  the  principal  value  and  is  the  one 
to  be  used  in  the  following  examples: 

1.  If  arcsin  i  =  0,  find  cot  6. 


If  arcsin  i  =  6,  then  sin  0  =  J.     To  construct  6,  draw  a 
*  The  symbol  sin~*  a  is  also  commonly  used  for  this  purpose. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


37 


right  angle  and  on  one  arm  lay  off  MP  =  1 ;  with  P  as  centre 
and  radius  3  draw  an  arc  cutting  the  other  arm  at  0.  Then 
MOP  =  e  and  OM  =  i/9  -  1  =  21/2^  Hence  cot  (9  =  2i/2. 

2.  If  arccot  (—  i)  =  6,  find  sin  6. 

Since  arccot  (—  i)  =6,  then  cot  d  =  —  \  and  0  must  be  an 


obtuse  angle.  To  construct  6,  lay  off  OM  =  —  1  and  at  M 
erect  MP  =  2.  Then  XOP  =  d  and  OP  =  i/5.  Hence  sin  (9 
=  2/1/5. 

34.  Exercise  IX. 

1.  Construct  arccos  (—  |),  arctan  |,  arcsin  (—  i). 

2.  Prove  arcsin  f  =  arctan  2/i/21. 

3.  Prove  arccos  (-  f)  =  arctan  (-  i/33/4). 

4.  Find  tan  (arcsin  |);  cos  [arccos  (—  i)]. 

5.  If  arccos  {2x^  -  2x)  =  27r/3,  find  x. 

6.  Find  arcsin  0,  arctan  1,  arccos  ^,  arctan  go  ,  arcsin  (— 4l/^), 
arccos  1. 

35.  Orthogonal  Projection.  AB  being  any  line  segment 
and  PQ  any  line,  draw  A  A'  and  BB'  perpendicular  to  PQ, 
Then  A'B'  is  the  orthogonal  projection  of  AS  on  PQ.  Through 
A  draw  AE  parallel  to  PQ  and  produce  BA  to  meet  PQ  in  L. 
Then  J^AS  =  B'LB  =  d,  and  AE  =  A'B'.  Since  AE-MB  = 
cos  6  we  have  AE  =  AS  cos  ^  and  therefore  A'B'  =  AB  cos  6- 
Hence  the  projection  of  any  segment  AB  on  a  line  is  AB  times 
the  cosine  of  the  angle  between  the  lines. 


38 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


Project  the  sides  of  the  triangle  ABC  (in  Fig.  33)  on  a  line 
PQ.     Since  A'B'+B'C'=  A'C  and  A'C'-^  C'B'=  A'B'  we  see 


4 


A' 

Fig.  32. 


B' 


that  the  sum  of  the  projections  on  any  line  of  two  sides  of  a 
triangle  is  equal  to  the  projection  on  that  line  of  the  third  side. 


36.  Functions  of  the  Sum  or  Difference  of  Two  Angles. 
To  prove 

sin  {A  +  B)  =  siaA  cos  B  +  cos  A  sin  By 

cos  {A-{-B)  =  cos  i4  cos  5  —  sin  i4  sin  B, 

The  following  proofs  are  valid  no  matter  what  may  be  the 
size  of  A  and  B.  For  the  student  to  see  this,  however,  is 
harder  than  to  limit  the  present  proofs  to  the  case  where  A 
and  B  are  acute  and  later  to  show  that  they  may  be  extended 
to  cover  all  cases. 

First  Method.  The  following  proofs  apply  whether  A-\-B 
is  acute  or  obtuse. 

Draw  XOX' =  A  and  X'OP  =  S.  Then  XOP  =  A+B.  From 
P  draw  PQR  perpendicular  to  0X\ 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


39 


By  Art.  35,  the  projection  on  any  line  of  OP  is  equal  to  the 
sum  of  the  projections  on  that  line  of  OQ  and  QP.  Projecting 
on  OX  gives 

r 


But 
Hence 


OP  cos  (A+B)  =  0Q  cos  A  +  QP  cos  XRP, 

cos  XRP  =  cos  (90°+ A)  =  -  sin  A. 

OP  cos  (A +5)  =00  cos  A-QP  sin  A. 
Dividing  by  OP, 

cos  {A-^B)=OQlOP  cos  A-QPJOP  sin  A, 
and  therefore 

cos  (A+P)  =  cos  B  cos  A  — sin  B  sin  A.     Q.  E.  D. 
In  the   second   figure,   projecting  OP  on  OF  gives 
OP  cos  [(A +5) -90°], 
which  is  equal  to  OP  cos  [90°  —  (A +P)].     Hence  in  either  figure, 
projecting  OP,  OQ  and  QP  on  OF  gives 
OP  cos  [90°-(A+B)]  =  OQ  cos  (90°- A)+OP  cos  {XRQ-90°) 

Hence 

OP  sin  {A+B)=OQ  sin  A+QP  cos  A, 

and  dividing  by  OP 

sin  (A+P)  =  cos  B  sin  A+sin  P  cos  A.       Q.  E.  D. 

Second    Method.     Draw  XOX''  =  A  and  X''OP  =  P;  then 

XOP  =  A  -\-  B.     To  get   triangles  of  reference  for  A,  P  and 

A  +  P,  take  any  point  P  on  the  terminal  line  of  A  +  P  and 


40 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


draw  PM  perpendicular  to  OX.     P  being  also  on  the  terminal 
line  of  B,  draw  PQ  perpendicular  to  OX";  and  Q  being  on  the 


Fig.  35. 

terminal  line  of  A  draw  QH  perpendicular  to  OX.     Through 
Q  draw  KQX'  parallel  to  OX.     Then  X'QP  is  90°+A. 

.     ,.  ,  p.      MP      MK-\-KP      HQ  ,  KP 
sm(A+B)=^  =  -^^-=^  +  ^. 

But  HQ  =  OQ  sin  A,   KP  =  QP  sin  (90°+A)=QP  cos  A. 


.'.  sin  (A+B)  =sin  A  ~p  +  cos  A 


QP 
OP 


=  sin  A  cos  B+cos  A  sin  B.     Q.  E.  D. 

Ai  rA^m     ^^      OH+HM      OH  ,  QK 

Also  cos  (A+5)=  ^-^-  =  -^^  -OP^OP' 

But  OH  =  OQ  cos  A,   QK  =  QP  cos  (90°+A)  =  -QP  sin  A. 

.  .  cos  {A-f-B)  =  cos  A  Yyp  —  sm  A  ^yp^ 

=  cos  A  cos  B  — sin  A  sin  B.     Q.  E.  D. 
37.  To  prove 

sin  (i4  —  5)  =  sin  A  cos  B  —  cos  A  sin  5 
cos  {A  —  B)  =  cos  A  cos  ^  +  sin  A  sin  5. 

Suppose  J5+5'  =  90°,  so  that  B'  also  is  acute. 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  41 

Then 

sin  (A-5)=sin(A+5'-90°)  =  -sin(90-A-BO 
=  -cos  {A+B') 
=  —cos  A  cos  B'+sin  A  sin  B' 
=  -cos  A  cos  (90°-B)+sin  A  sin  (90° -5) 
=  —cos  A  sin  B  +  sin  A  cos  B.  Q.  E.  D. 

Similarly 

cos  {A  -  B)  =  cos  (A  +  B'  -  90°) 
=  cos  (90  -  A  -  B') 
=  sin  (A  +  BO 
=  sin  A  cos  5'  +  cos  A  sin  5' 
=  sin  A  cos  (90°  -  5)  +  cos  A  sin  (90°  -  B) 
=  sin  A  sin  5  +  cos  A  cos  B.  Q.  E.  D. 

38.  That  these  formulae  are  true  when  A  and  B  are  not 
acute  may  be  shown  in  any  special  case  as  follows : 

Suppose  A  to  be  an  angle  in  the  third  quadrant  and  B  to 
be  one  in  the  second.  Then  A  can  be  expressed  in  the  form 
180°  +  A'  where  A'  will  be  an  acute  angle.  Similarly  B  =  90° 
+  B\  B'  being  acute. 

Then 

sin  (A  +  5)  =  sin  (270°  +  A'  +  B') 

=  -  cos  {A'  +  B')  §  31. 

=  —  cos  A'  cos  B'  +  sin  A'  sin  B' 
=  -  cos  (A  -  180°)  cos  {B  -  90°) 
+  sin  (A  -  180°)  sin  (5  -  90°) 
=  cos  A  sin  J5  +  sin  A  cos  B. 
In  like  manner 

cos  {A  -  B)  =  cos  (90°  ■\-  A'  -  B') 
=  -  sin  (A'  -  BO 
=  —  sin  A'  cos  B'  +  cos  A'  sin  B' 
=  -  sin  (A  -  180°)  cos  {B  -  90°) 
+  cos  (A  -  180°)  sin  {B  -  90°) 
=  sin  A  sin  B  +  cos  A  cos  B. 

39.  From  the  preceding  formulae  we  get 
sin  (A  +  B)      sin  A  cos  B  +  cos  A  sin  B 


tan  (A  +  B)  = 


cos  (A  +  B)   cos  A  cos  B  —  sin  A  sin  B' 


42  ELEMENTS   OF  PLANE   TRIGONOMETRY. 

Dividing  numerator  and  denominator  by  cos  A  cos  B,  we 
obtain 

sin  A  cos  B      cos  A  sin  B 
cos  A  cos  -B      cos  A  cos  B 


cos  A  cos  B      sin  A  sin  5 

.-.  tan  (A  +  5)  = 


cos  A  COS  5      COS  A  cos  5 
tan  A  4-  tan  ^ 


I  —  tani4tan5 
Similarly  it  can  be  proved  that 

...       o\        tan  i4  —  tan  B 

tan  (i4  -  5)  = 


I   +  tan  A  tan  B' 

40.  Exercise  X. 

Apply  the  formulae  of  §§  36-39  to  the  following  examples, 
simplifying  when  possible : 

1.  sin  (180°  -  d),  2.  cos  (90°  +  6). 

3.  tan  (180°  +  6).  4.  sin  (2A  +  B). 

5.  cos  (90°  -  e).  6.  tan  (0°  -  6). 

7.  By  expanding  cos  (6  —  6)  find  cos  0°. 

8.  Similarly  find  sin  0°  and  tan  0°. 

9.  From  tan  (45°  +  45°)  find  tan  90°. 

10.  From  sin  (45°  +  30°)  find  sin  75°. 

11.  Find  sin  15°  and  cos  15°. 

12.  If    sin  A   =  f    and    cos    B  =  A    find    sin    (A  +  B), 
cos  (A  —  B)  and  tan  (A  +  J5),  A  and  B  being  acute. 

13.  If   cos   A   =  —  f   and   sin   B  =  \   find    cos    (A  +  B), 
sin  (A  —  B),  and  tan  {A  —  B),  A  and  B  being  obtuse. 

14.  If  A  =  arcsin  \  and  B  =  arccos  (—  f)  find  sin  (A  —  B). 

15.  Prove  the  formula  for  sin  (A  —  B)  when  A  and  B  are 
in  the  third  quadrant. 

16.  Prove  the  formula  for  cos  (A  +  B)  when  A  is  in  the 
second  quadrant  and  B  in  the  fourth. 

41.  Functions  of  an  Angle  in  Terms  of  Half  the  Angle. 

sin  26  =  sin  {B  -\-  6)  =  sin  6  cos  6  -\-  cos  d  sin  d 

.'.  sin  26  =  2  sin  6  cos  B. 

cos  2B  =  cos  {B  +  B)  =  cos2  ^  -  sin2  6. 

,         -         2  tan  ^ 

tan  2B  = 7 — — -. 

I  -  tan2  B 


ELEMENTS 

I  OF  PLANE 

TRIGONOMETRY. 

Similarly 

sin  d  = 

^^^(2  +  2)  = 

=  2 

.  e 

sm  -  cos 

e 

'2 

cos^  = 

a 

2  tan- 
Z 

1  -  tan^l 

e 
2 

tan^  = 

43 


42.  Functions  of  an  Angle  in  Terms  of  Double  the  Angle. 

Since     cos-  6  —  sin^  6  =  cos  26     and     cos^  6  +  sin^  6=1 
we  get  by  adding  and  subtracting 

2  cos2  0  =  1+  cos  26    and     2  sin^  ^  =  i  —  cos  26, 
From  cos^  -  —  sin^  -  =  cos  6      and      cos^  ^  +  sin^  ^  =  1 
we  can  get  in  the  same  way 

a  fi 

2  cos^  -  =  1  +  COS  0     and     2  sin^  -  =  1  —  cos  ^. 

z  z 

Hence  cos|=-JL±^^; 

2  \        2       ' 

sm-  , 

1  ,       <9  2        .        1  -  cos  ^ 

and  tan-  =  .      1  u 


-4\ 


2     ^„  e        \i  +  cosfl 

cos  2 

43.  It  is  well  to  be  able  to  state  any  of  these  formulae  in 
words.  For  example,  the  first  one  in  Art.  41  is:  the  sine  of 
any  angle  is  twice  the  sine  of  half  the  angle  times  the  cosine 
of  half  the  angle.     This  enables  us  to  write 

sm  0  =  2  sm  ^  cos  ^, 
z       z 

'     rA  ^    n^       ^    •    A+  B        A+B     . 
sm  {A  +  B)  =  2  sm  — ^r —  cos  — ^r — ,  etc. 


44 


ELEMENTS  OF  PLANE   TRIGONOMETRY. 


44.  Factoring  Formulae.     To  prove 

sin^  +  sm9  =  2  sin  cos -, 

sin  0  -  sin  0  =  2  cos ^sin ^, 

2  2     ' 

cos  0  +  COS  0  =  2  cos^ ^cos ~y 


,  e  ^-  4>  .  e  -  4> 

COS  B  —  cos  0  =  —  2  sin —     sin^ ~, 

22 

From    sin  {A  -\-  B)  =  sin  A  cos  B  +  cos  A  sin  B 

and  sin  (A  —  5)  =  sin  A  cos  5  —  cos  A  sin  B 

we  get      sin  {A  -\-  B)  -\-  sin  (A  —  B)  =  2  sin  A  cos  5 

and  sin  {A  -{-  B)  —  sin  (A  —  5)  =  2  cos  A  sin  5. 

Similarly 

cos  (A  +  B)  +  cos  (A  —  J5)  =  2  cos  A  cos  B, 
cos  (A  +  B)  —  cos  (A  —  B)  =  —  2  sin  A  sin  B. 

Put        A  +  B  =  ^, 
A  -  B  =  0 

Then  2A  =  ^  +  0     and     A 


(a) 
(^) 

(c) 


^  +  0 


2B  =  61  -  0    and     B  =  ^-2^. 

Using  these  values  for  A  and  B  in  (a),  (6),  (c)  and  {d)  we 
get  the  desired  formulae. 
45.  Exercise  XI. 


1.  Given  arctan  (—  f)  =  ^  find  sin  612.     Since  arctan  (—  f) 
6,  tan  e  =  —  \  and  B  must  be  in  the  second  quadrant. 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  45 

Cut  off  OM  =  -  4:  and  draw  MP  =  3.  Then  XOP  =  d  and 
OP  =  5.  From  Art.  42,  2  siri^  (9/2  =  1  -  cos  ^  =  1  -  (-  |) 
=  f.  Hence  sin  ^/2  =  ±  1/^%;  but  since  0  lies  between  90° 
and  180°,  6/2  lies  between  45°  and  90°  and  the  positive  sign 
must  be  taken.     So  sin  6/2  =  Vju  =  AV  10. 

2.  Given  6  =  arcsin  ( -  |)  find  sin  26,  cos  6/2,  sin  6/2. 

3.  If  A  =  arctan  f  find  sin  2 A  and  tan  2 A. 

4.  By  aid  of  Art.  42  find  the  functions  of  22°  30'. 

5.  Given  cos  6  =  —  j,  6  being  in  the  third  quadrant,  find 
cos  6/2,  tan  6/2,  tan  26. 

6.  If  arctan  ^  =  A  and  arcsin  (—  |)  =  B,  find  sin  (2A  +  B) 
and  cos  {2 A  —  B). 

7.  If  arccot  (—  J2)  =  ^j  find  tan  A/2,  sin  A/2  and  tan  2A. 

8.  If  sin  A  =  If  and  cos  B  =  —  f ,  A  and  B  being  in  the 
second  quadrant,  find  tan  (A  —  B),  cos  A/2,  tan  B/2. 
Factor:  9.  sin  5 A  +  sin  A.  10.  cos  3 A  +  cos  A. 

11.  sin  80°  -  sin  70°.  12.  cos  40°  -  cos  10°. 

13.  cos  (A  +  J5)  +  cos  (A  -  B). 

14.  sin  (3A  +  B)  -  sin  (A  +  SB). 

15.  sin  55°  +  cos  65°.  16.  cos  15°  -  sin  35° 

17.  cos  {26  +  <^)  +  cos  (2(9  -  0). 

18.  cos  A  —  cos  3A. 

,^     .    SA+B       .    A  +3J5 

19.  sm  — - — -  —  sm  — . 

20.  sin  (A  +  5  -  C)  -  sin  (A  -  B  +  C). 

21.  cos  (A  +  5/4)  +  cos  (A/2  +  35/4). 

22.  If  a  +  j8  =  45°  prove  that 

(1  +  tana)(l  +  tan|8)  =  2; 
also  that 
2  sin  a  cos  a  =  (cos  jS  +  sin  /8)(cos  jS  —  sin  /3). 

23.  If  tan  6  =  '^  and  cot  0  =  V  prove  that 

tan  (20  +  0)  =  |. 

24.  If  a:  =  arcsin  l/i/5  and  cot  1/  =  3  prove  that 

a:  +  2/  =  45°. 
46.  Identical  Equations. 
1.  Prove 

cos  A   cos  15 


46  ELEMENTS  OF   PLANE   TRIGONOMETRY. 

Changing  to  sines  and  cosines 

sin  A      sin  B  _  sin  {A  +  B) 
cos  A      cos  B  ~~  cos  A  cos  B' 
Clearing  of  fractions 

sin  A  cos  B  +  cos  A  sin  B  =  sin  (A  +  B). 

But  this  is  true  by  Art.  36;  and  since  the  steps  may  be 
reversed  the  original  relation  is  also  true. 

2.  Prove  tan  26  =  tsLU  6  +  tan  0  sec  2d 
Changing  to  sines  and  cosines 

sin  2d  _  sin  ^      sin  6      1 
cos  20  ~  cos  6      cos  0  cos  20' 

Clearing  of  fractions 

sin  20  cos  ^  =  sin  0  cos  20  +  sin  0, 
Transposing 

sin  20  cos  0  —sin  0  cos  20  =  sin  6 

But  the  left  side  is  sin  (20  —  0).    Hence  the  identity  has  been 
proved. 

3.  Prove 

sin  50-2  sin  3(9  +  sin  ^      ^      ^^ 

^ ^ iTT— i a  —  tan  66. 

cos  50  —  2  cos  S0  +  cos  0 

The  left  side  is  equal  to 

(sin  50  +  sin  0)  -  2  sin  3(9  ^  2  sin  30  cos  2(9-2  sin  30 
(cos  50  +  cos  0)  -  2  cos  30  ~  2  cos  30  cos  20-2  cos  30 

_  2  sin  30  [cos  20  -  1] 
-  2  cos  30  [cos  20  -  1] 

sin  30        ^         o^        ^      -n      -n. 

= ^  =  tan  30.     Q.   E.   D. 

cos  30  ^ 

4.  If  A,  B,  and  C  are  the  angles  of  a  triangle  prove 

ABC 
sin  A  +  sin  B  +  sin  C  =  4  cos  ^  cos  ^  cos  ^. 


A  +  B, 
2      ^ 

C 
2 

^90°, 

sin 

2       ~ 

cos  -  and  cos 

A  +B 
2 

^sm^. 

(sin  A  + 

sin 

B)  + 

sin 

C  =  2  sin 

A+B       A- 

2      ^^^     2 

\-2  sm  2  cos  2 

ELEMENTS   OF   PLANE   TRIGONOMETRY.  47 

^        C       A-B.^       A+B       C 
=  2  cos  -  cos  — h  2  cos  — - —  cos  ^ 

^        C\      A  -  B  ^         A  +  Bl 

s  2  cos  2  [cos  — h  cos  — 2 —  I 

=  2  COS  2  [2  cos  -  cos  -J     (by  §  44) 

ABC  r^     ^     r^ 

=  4  cos  "2  cos  2  cos  -^.  Q.  E.  D. 

47.  Exercise  XII. 

Prove  the  following  identities: 

1.  tan  A  —  tan  B  =  sin  {A  —  B)/cos  A  cos  B. 

2.  sin  (A  +  B)  cos  (A  -  B) 

+  cos  (A  +  B)  sin  (A  -  B)  =  sin  2  A. 

3.  sin  (A  +  J5)  sin  (A  -  B)  =  sin^  A  -  sin^  B. 

4.  cos  (n  +  1)  A  cos  A  +  sin  (n  +  1)  A  sin  A  =  cos  nA. 

5.  cos  (A  +  5)  cos  {A  -  B)  =  cos^  A  -  sin^  B. 
tan(A  +  B)+tan(A-B)_ 

^*    1  -  tan  (A  +  5)  tan  (A  -  5) "  ^^"^  '^^• 

7.  sin  ?i^  cos  ^  =  sin  (n  +  1)^  —  cos  nO  sin  d. 

8.  tan  2a;  +  sec  2x  =  (cos  a;  +  sin  a;)/(cos  x  —  sin  x). 

„    sin  3A  +  sin  2A  +  sin  A       ^      ^  . 

9.  5-7—1 ^sr:r~i A  —  tan  2A. 

cos  3A  +  cos  2A  +  cos  A 

10.  (cot  e  +  l)/(cot  0  -  1)  =  (1  +  sin  26/)/cos  20. 

11.  sin  30/sin  6  —  cos  30/cos  0  =  2. 

12.  2  sin  3A  cos  A  =  sin  4A  +  sin  2A. 

13.  cos  A  —  cos  3A  =  2  sin  2A  sin  A. 

14.  cos  30  =  2  cos  40  cos  0  —  cos  50. 

^  p    sin  8A  —  sin  6A  +  sin  4A  —  sin  2A  .  _  . 

1^-  E71 TTi — , n r.  A  —  ~  cot  oA. 

cos  8A  —  cos  6A  +  cos  4A  —  cos  2A 

16.  sin  20  =  2  tan  0/(1  +  tan^  0). 

17.  cos  0  =  (1  -  tan2  0/2)/(l  +  tan^  0/2). 

18.  (tan  A  +  tan  B)/(cot  A  +  cot  B)  =  tan  A  tan  B. 

19.  (tan  A  -  tan  5)/(cot  A  +  tan  B)  =  tan  A  tan  {A  -  B). 

20.  1    +  tan  A  tan  5  =  cos  (A  —  J5)/cos  A  cos  B. 

21.  tan  (A  +  B)  tan  (^  -  B)  ^  T-L^TCb- 

22.  tan  (7r/4  +  0)  =  (l  +  tan  0)/(l  -  tan  0). 

23.  cot  (7r/4  +  0)  =  (cot  0  -  l)/(cot  0  +  1). 


48  ELEMENTS  OF  PLANE   TRIGONOMETRY. 

24.  tan2  (7r/4  -  A)  =  (1  -  sin  2A)/(1  +  sin  2A). 

25.  1  +  tan  2e  tan  6  =  sec  20. 

26.  cos2  A  +  cos2  (A  +  60°)  +  cos^  (A  -  60°)  =  3/2. 
27  tang  cot  0         ^ 

*  tan  0  -  tan  30  "•"  cot0  -  cot30  ~    * 

^^*  tan  30  +  tan  d  ~  cot  30  +  cot  6  ~  ^^*  ^"^^ 

29.  sin  30  cos  0  +  cos  50  sin  30  =  cos  30  sin  0  +  sin  50  cos  30. 

30.  2  sin  (7r/4  +  A)  sin  (x/4  -  A)  =  I  -  vers  2A. 

31.  (cos  20  +  cos  60)/(vers  60  -  vers  20)  =  cot  20  esc  40  -  1. 

32.  1  -  vers  20  =  (cot^  0  -  l)/(cot2  0+1). 

„„    2  +  sin  0  -  vers  0  ^  0 

33.  r— ^-, —  =  cot  -. 

sm  0  +  vers  0  2 

vers  70  —  vers  0  _  cos  0  +  cos  70  _  ^ 

vers  50  —  vers  30      cos  30  +  cos  50  ~" 

„^    vers   20  vers  0      ^      ^  ^      0 

35-  •    ozi   •    ^     -  tan  0  tan  -. 

sm  20  sm  0  2 

A  +  J5  A  —  B 

36.  tan  — ^ —  —  tan  — ^—  =  2  sin  B/(cos  A  +  cos  5). 

^^    sin  (g  -  i8)      sin  (j(3  -  7)    |  ■  sin  (7  -  «)  _  ^ 

sin  a  sin  |8        sin  j3  sin  7         sin  7  sin  o: 
38.  tan  q;/(1  —  cot  2a  tan  «)  =  sin  2a. 

sin  (g  +  /3)  +  sin  (a  —  jg)  ^  tan^ 

sin  (a  +  /3)  —  sin  (a  —  /3)  ~  tan  jS  * 
In  examples  40^9  A,  B,  and  C  are  the  angles  of  a  triangle. 

40.  sin  (A  +  5)  -  sin  C. 

41.  cos  (A  +  5)  =  -  cos  C. 

,^     .    A  +  B  C 

42.  sm  — z —  =  cos  - . 

B+  C        .A 

43.  cos  — - — -  =  sm  ^  . 

44.  sin  2 A  +  sin  25  +  sin  2C  =  4  sin  A  sin  B  sin  C. 

ABC 

45.  sin  A  +  sin  5  —  sin  C  =  4  sin  --  sin  -  cos  -  . 

ABC 

46.  cos  A  +  cos  5  +  cos  C  =  4  sin  ^  sin  ^  sin  -  +  1. 

47.  tan  A  +  tan  B  +  tan  C  =  tan  A  tan  B  tan  C. 

48.  tan  —  tan  -  +  tan  ^  tan  -  +  tan  ^  tan  —  =  1. 


ELEMENTS   OF  PLANE  TRIGONOMETRY.  49 

49.  If  sin  A  =  2  cos  B  sin  C  prove  that  the  triangle  is  isosceles. 

48.  Conditional  Equations. 

Throughout  the  work  equalities  have  been  met  with  which 
are  true  for  any  value  of  the  angles  involved.  These  have 
varied  from  simple  relations  like  tan  d  =  sin  0/cos  6  and  sin^  6 
+  cos^  6  =  1  to  the  more  complicated  ones  of  the  previous 
article.  They  are  identical  equations  or  more  briefly  identities. 
Equalities  which  are  true  only  for  special  values  of  the  angles 
are  conditional  equations,  often  called  merely  equations.  To 
solve  these  is  the  present  problem. 

Find  all  the  values  of  6  between  0°  and  360°  which  satisfy: 

1.  sin2  0  +  cos  ^  =  1. 
Changing  into  one  function  we  get 

1  —  cos^  6  +  cos  ^  =  1, 
—  cos^  6  +  cos  ^  =  0, 
-  cos  ^(cos  0  -  1)  =  0. 
.*.  cos  6  =  0,     and    6  =  90°  or  270°, 
or  cos  ^  =  1,     and     6  =  0°. 

2.  2  cos  0  =  sin  6  +  1. 

Substituting  for  cos  6  its  value  in  terms  of  sin  6 
2i/l  -  sin2  (9  =  sin  0  +  1, 
4(1  -  sin2  6)  =  sin2  6  +  2  sin  6  -\- 1, 
5  sin2  ^  +  2  sin  ^  -  3  =  0, 
(5sin^  -  3)(sin(9  +  1)  =  0, 
.-.  sin  ^  =  -  1     and     6  =  270°, 
or         sin  <9  =  f     and    6  =  36°  52'  12''  or  143°  7'  48". 

Since  both  sides  of  this  equation  were  squared  extraneous 
roots  were  introduced.  It  is  necessary  to  test  the  angles  found 
to  see  which  suit  the  original  equation.  It  is  seen  immediately 
that  143°  7'  48"  does  not  satisfy  the  equation,  the  left  side 
being  negative  and  the  right  positive  in  that  case. 

3.  sin  26  cos  6  =  sin  6 
Hence 

2  sin  6  cos^  0  =  sin  6, 


315°. 


.*.  sin  0  =  0     and     6  =  0° 
or 

or     180°, 

2  cos2  (9  =  1,  cos  0  =   ±  l/v/2  and  6  = 
4.  cos  36  +  cos  0  =  sin  3  ^  +  sin  6. 
4 

45°,  135° 

50  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

By  Art.  44 

2  cos  2d  cos  ^  =  2  sin  26  cos  0. 
.'.  COS0  =  0     and    d  =  90°     or    270°, 
or  cos  26  =  sin  26,     that  is,     tan  26  =  1. 

Hence  26  =  45°,  225°,  405°,  585°. 

6  =  22°  30',  112°  30',  202°  30',  292°  30'. 

The  above  examples  illustrate  some  of  the  methods  of  solving 
trigonometric  equations.  In  all  of  them  b}^  factoring  the 
equation  values  of  some  function  of  an  angle  are  obtained. 
In  the  first  two,  everything  is  expressed  in  terms  of  a  single 
function.  This  is  the  most  common  method.  Having  in  any 
way  found  the  value  of  one  function  it  remains  to  find  the  values 
of  the  angle.  For  this  it  is  very  useful  to  remember  that  any 
function  of  6  has  the  same  numerical  value  as  the  same  function 
of  180°  ^  6  and  360°  —  6,  two  of  the  four  angles  giving  positive 
values,  the  remaining  two  negative  ones.  Thus  if  cos  ^  =  —  ^, 
we  look  for  the  acute  angle  whose  cosine  is  ^.  This  is  60°. 
Since  cos  6  is  negative,  6  must  be  in  the  second  or  third  quadrants; 
it  is  therefore  180°  ±  60°,  that  is,  120°  or  240°. 

The  student  must  guard  against  cancelling  out  a  factor 
without  keeping  account  of  the  corresponding  roots.  Thus 
in  example  3  a  common  error  is  to  merely  cancel  out  sin  6 
and  reduce  the  equation  to  2  cos^  ^  =  1.  Also  if  it  be  found 
necessary  to  square  both  sides  of  an  equation  A  =  B,  thus 
obtaining  A^  =  B^,  the  solutions  will  include  those  oi  A  = 
—  B,  in  addition  to  the  ones  desired.  In  this  case  the  angles 
should  be  substituted  in  the  original  equation  and  those  dis- 
carded which  do  not  satisfy  it.  In  example  (2)  an  instance  of 
this  was  shown. 

49.  Exercise  XIII.  Find  all  values  of  0  between  0°  and 
360°  which  satisfy: 

1.  cos  0  +  sec  ^  =  |. 

2.  sin  ^  =  1  —  vers  6. 

3.  sin  6  =  tan  6. 

4.  2  cos  ^  =  3  tan  6. 

5.  tan2  ^  +  3  cot^  6  =  4. 
6.s3  COS0  =  2  sin2^. 

7.  COS  26  +  sin2  0  =  f . 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  51 

.  8.  sin  5^  +  sin  Sd  =  cos  0. 
9.  sin  2^  =  3  tan  ^  -  3  tan  0  vers  2d. 

10.  sin  6(9  =  sin  40  -  sin  20. 

11.  cos  3^  +  cos  20  +  cos  0  =  0. 

12.  cot  2^  +  tan  ^  +  1  =  0. 

13.  sin  20  +  sin  <?  =  cos  20  +  cos  0. 

14.  2  sec  ^  =  tan  0  +  cot  0. 

15.  6  tan2  (9-4  sin^  0  =  1. 

16.  -^=2  +  cot^. 
vers  0 

17.  tan  ^  -  cot  0  =  cot  2(9. 

18.  sin  3^  -  sin  ^  =  cos  3^  +  cos  0. 

a 

19.  sin  0  +  sin  20  =  i/3 cos- . 

20.  sin  0  +  sin  20  +  sin  30  =  1  +  cos  (9  +  cos  20. 

21.  sin  0  -  cos  0  =  V2, 

22.  vers  0  =  vers  20. 

23.  sec2  0  +  csc2  0  =  ^^.  , 

24.  tan  0  +  2  cot  0  =  I  esc  0. 

25.  2  sin  30  cos  0  =  2  sin  0  cos  30-1. 

26.  tan  20-2  vers  0  +  2  =  0. 

27.  cos  20  +  sin  0  =  2  vers  20. 

28.  sin  20  =  2  sin  0  +  vers  0. 

Are  the  following  identical  or  conditional  equations?     If  the 
former,  prove  the  identities;  if  the  latter,  solve  the  equations. 

29.  sin  0  +  CSC  0  =  |. 

30.  tan  2x  —.tan  x  =  tan  x  sec  2x. 

31.  cot  A  —  cot  2A  =  CSC  2A. 

32.  sin  2A  =  2  sin^  A. 

33.  sin  30  +  sin  20  +  sin  0  =  0. 

34.  cos2  0  =  1  -  sin  0. 
cot  20 


35. 

cot  20  +  tan0~  ^^ 

36. 

cos  A  +  cos  3 A  +  cos  5A  +  cos  7 A 

=  4  cos  A  cos  2A  cos  4A 

37. 

vers  20  =  2  sin  0. 

38. 

cos  (0  +  7r/4) 

-■    ;^    — 77^  =  sec  20  -  tan  20. 

cos  (0  —  7r/4) 

52  ELEMENTS   OF  PLANE  TRIGONOMETRY. 

39.  cot  20  =  tsLnd  -  1. 

40.  cot2  ^  (2  CSC  6*  -  3)  +  3  (esc  6  -  1)  =  0. 

41.  3  tan2  (9  +  8  cos^  6  =  7. 

42.  tan  d  -\-  cot  6  =  4. 

43.  sin  0+1/3  cos  (9  =  1. 


CHAPTER  IV. 

50.  Solution  of  Triangles.  In  geometry  it  is  shown  that  a 
triangle  can  be  constructed  when  any  three  of  its  parts  are  given, 
provided  that  one  of  the  three  be  a  side.     Four  cases  arise: 

1.  One  side  and  two  angles; 

2.  Two  sides  and  an  angle  opposite  one  of  them; 

3.  Two  sides  and  the  included  angle; 

4.  Three  sides. 

In  the  second  case  two  triangles  may  sometimes  be  found 
having  the  required  parts. 

We  proceed  to  derive  formulae  for  computing  the  unknown 
parts. 

51.  Law  of  Sines.  The  sines  of  the  angles  of  any  triangle 
are  proportional  to  the  opposite  sides. 


First  proof.     In  the  triangle  ABC  draw  BD  perpendicular 
to  AC.     Then  in  both  figures 

.      .       DB  ,     DB        .    ^ 

sm  A  = ,     and     —  =  sm  C. 

c  a 

Multiplying  one  of  these  equations  by  the  other  and  cancelling 
DB,  we  get 

sin  A  _  sin  C 
a  c 

In  like  manner  by  dropping  from  C  a  perpendicular  to  AB, 
it  may  be  proved  that 

53 


54 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


Hence 


sin  A      sin  B 
a      ~     b    ' 

sin  A 

= 
a 

sm  B      sin  C 
-      6     -     c    • 

Second    vn 

oof.      Circumscribe 

Fig.  38. 


circle  about  ABC  and  draw  the 
diameter  through  A.  Join  BD. 
Then,  since  the  angle  C  is  equal  to- 
the  angle  D,  sin  C  =  sin  D  =  c/AD 
=  c/d,  where  d  is  the  diameter  of 
the  circumcircle.     Hence 

sin  C  _  1 
c         d' 


Similarly 

sm  A      1          ,     sm  B      1 

—  3     and        ,       =  J. 

ad                  b          d 

Therefore 

sin  A      sin  B      sin  (7      1 
a             b      ~~     c         d' 

52.  Since  sin  A /sin  5  =  a/b,  we  have  by  composition  and 

division 

A-B       A+B 

—  cos — 7^ — 


Similarly 


sin  A  —  sin  B 
sin  A  +  sin  5 

••  a+  6 

6-  c 
6+  c 


2  sin 


^       A  -B  .    A  +  5' 
2  cos  —  - —  sm  — ^ — 


and 


tan  i(A  -  B) 
tan  i(A  +  By 

tan  i(Jg  -  C) 
tan  -1(5  +  C)' 


c^^  ^  tan  i(C  -  A) 

c-]-  a     tanKC  +  A)* 

53.  Law  of  Cosines.  The  square  on  one  side  of  any  triangle 
is  equal  to  the  sum  of  the  squares  on  the  other  two  sides  di- 
minished by  twice  the  product  of  these  sides  times  the  cosine 
of  the  angle  between  them. 

To  prove  a^  =  ¥  +  c^  —  2bc  cos  A. 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


55 


Two  cases  arise,  one  when  A  is  acute  and  the  other  when  A 
is  obtuse.     In  each,  from  C  draw  CD  perpendicular  to  AB. 
a^  =  DC^  +  DB\ 
DB  =  DA  +  AB  =  AB  -  AD  =  c  -  AD. 


Hence 

a2  ='DC'  +  (c  -  ADy  =  DC'  +  c^-  2c'AD  +  AD^ 
=  ¥  +  c^  -2c'AD. 
But         AD  =  h  cos  A  (being  negative  when  A  is  obtuse") 
.*.  a^  =  ¥  +  c^  -  2hc  cos  A. 

0^  — |—  C^  —  d^ 

This  may  be  written    cos  A  = 
Similarly  cos  B  = 

and  cos  C  =         ^  , 

2ab 

54.  Law  of  Tangents.     These  equations  are  not  in  form  for 
logarithmic  work  but  from  them  we  can  get  others  that  are. 


26c 

* 

c^  +  a"  - 

62 

2ca 

) 

a2  +  62  _ 

-   C2 

tan 


2.  aIft 


—  cos  A 


¥  +  c"-  a^ 


26c 


cos 


^M 


62  +  c2  -  a2 


1  + 


26c 


-4 
'4 


r2  _ 


(62  -  26c  +  c2) 


Putting 
then 


(62  +  26c  +  c2)  -  a2 

(a  -  6  +  c)(a  +  6  -  c) 
(a  +  6  +  c)(6+  c- a)* 
a  +  6  +  c  =  2s, 
a  +  6  —  c  =  2(s  —  c),  etc. 


56  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

Hence  taiiTr  =  ^1- — .  .    ^^ 

2        \      s(s  -  a) 


.:  tan  i  =  ^^      (s-aKs-bKs-cl 
2      s  —  a  \  s 

Putting  (see  Art.  55) 

^  ^     lis  -  a){s  -  b){s  -  c) 


we  get 

Similarly 

and 


tan-  = 

2 

S 

r 

a' 

tan  —  — 

r 

5 

— 

6' 

2 

5 

r 

c 

55.  Radius  of  Inscribed  Circle.  Draw  the  bisectors  of  the 
angles  of  the  triangle  AB  C.  These  meet  at  0,  the  centre  of  the 
inscribed  circle;  from  0  drop  perpendiculars  to  the  sides  of 
the  triangle.     Since  BCi  =  BAi,  ACi  =  ABi  and  CAi  =  CBi, 


Fig.  40. 

we  have  BCi  +  ACi  +  CAi  =  BAi  +  ABi  +  CBi.  Each  side 
of  this  equation  must  therefore  be  half  the  sum  of  the  sides. 
Hence 

s  =  BCi  +  ACi  +  CAi  =  c  +  CAi,    or    CAi  =  s  -  c. 


But 


.       nrA         .      C      A,0       AxO 

tan  OCAi  =  tan  —  =  ^^-r-  = . 

2      CAi      s  —  c 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


57 


Comparing  this  with  the  last  equation  of  the  preceding  article 
shows  that  AiO  =  r.  Hence  the  radius  of  the  inscribed  circle 
is  given  by ■_ 

—      )(g  ~  ci){s  —  b){s  —  c) 

56.  Area.     Drop  a  perpendicular  from  B  on  the  opposite 

side,    produced    if    necessary. 
The  area  ABC  isib-DB;  but 

since 

DB       .    ^ 
- —  =  sm  C, 

a 

we  have  on  replacing  DB  by 
its  value 

area  =  ^ah  sin  C.       (1) 

This  formula  gives  the  area  in  terms  of  two  sides  and  the 

included  angle. 

From  the  law  of  sines, 

csmA 


Fig.  41. 


a  = 


and 


h  = 


Using  these  in  (1)  gives 

area  =i 


sinC 

csin  B 
sin  C 

&  sin  A  sin  B 


sinC 


(2) 


This  formula  may  be  used  when  one  side  and  two  angles  are 
given. 

Finally,  in  Fig.  40,  ABC  =  AOB  +  BOC  +  COA  =  icr  + 
\ar  +  ^hr  =  rs.  Using  the  value  of  r  obtained  in  the  preceding 
article  we  get 

area  =  t/s(s  —  a){s  —  h){s  —  c),  (3) 

a  formula  giving  the  area  in  terms  of  the  sides. 

57.  Case  I.  Given  One  Side  and  Two  Angles.  The  third 
angle  may  be  found  from  the  relation 

A-\-B  +  C  =  180°; 

the  remaining  sides  may  be  found  from  the  law  of  sines. 


58  ELEMENTS  OF  PLANE   TRIGONOMETRY. 

Example.     Solve  the  triangle  ABC  in  which 
B  =  100°  21'  10",     C  =  58°  17'  20''     and    a  =  31.656. 

f  A  =  180°  -  (5  +  C)  =  21°  21'  30", 

I 

Formulse  i    6  =  a  sin  B/sm  Aj 
t    c  =  a  sin  C/sin  A, 
Taking  logarithms  : 

log  a  =  11.50045  -  10 
-  log  sin  A  =    9.56134  -  10 

1.93911  1.93911 

+  log  sin  B*  =    9.99287  -  10      +  log  sin  C  =    9.92978  -  10 
log  h  =    1.93198  log  c  =    1.86889 

.-.  h  =  85.502  .-.  c  =  73.942 

Hence  the  solution  is 

A  =  21°  21'  30",  h  =  85.502,  c  =  73.942. 
58.  Case  II.  Given  Two  Sides  and  the  Angle  Opposite  One 
of  Them.  If  a,  h  and  B  are  known  A  may  be  found  from  the 
law  of  sines,  sin  A  =  a  sin  B/b.'\  But  since  it  is  sin  A  that  is 
given  by  this  equation,  A  may  be  either  an  acute  angle  or  its 
supplement,  and  in  general  there  will  be  two  values  of  A, 
Having  found  the  values  of  A,  C  may  be  found  from  A  +  B  +  C 
=  180°  and  finally  c  may  be  found  from  the  law  of  sines. 
Example.     Solve  the  triangle  in  which 

a  =  413.28,     h  =  378.19,     and    B  =  50°  16'  25". 
From  the  law  of  sines  sin  A  =  a  sin  B/b. 
log  a  =    2.61625 
+  log  sin  B  =    9.88599  -  10 
12.50224  -  10 
-log  6  =    2.57771 
log  sin  A  =    9.92453  -  10 
.-.  A  =  57°  11'  30"    or     122°  48'  30". 
Call  the  obtuse  angle  A'.     Then 

C  =  180  -  (A  +  J5),  C  =  180  -  (A'  +  B), 

=  72°  32'  5",  =  6°  55'  5". 

*SeeArt.31. 

t  If  a  sin  Blh>l  the  triangle  is  impossible  since  sin  A  cannot  be  greater 
than  1. 


ELEMENTS  OF   PLANE   TRIGONOMETRY. 


59 


From  the  law  of  sines 

6  sir 

C    =    -T— 

Sin 

,  _  5  sin  C 
~    sin  B 

log  h  =  12.57771 

-  10 

-  log  sin' B  =    9.88599 

-  10 

2.69172 

2.69172 

+  log  sin  C  =    9.97950  - 

-10 

+  log  sin  C 

=  9.08081  -  10 

log  c  =    2.67122 

logc' 

=  1.77253 

.-.  c  =  469.05 

.-.  c' 

=  59.229 

Hence  the  solutions  are 

A  =  57°  IV  30'' 

A' 

=  122°  48' 

30"] 

C  =  72°  32'    5" 

or 

C 

=      6°  55' 

5" 

c  =  469.05 

c' 

=  59.229. 

. 

Fig.  42. 


In  the  above  example  then,  there  are  two  triangles  which  satisfy 

the  conditions.     The  solution  of  every  problem  under  this  case 

proceeds  as  this  one  did 

till  we  try  to  get  C     In 

certain  examples  A'  -\-  B 

will  be  found  to  be  greater 

than  180°.     It  is  evidently 

impossible   to    find  C  in 

that  case  and  ther0>  is  only 

one    solution.      This    will 

occur  when  the  angle  given 

is  opposite  the  larger  of  the  sides  given. 

To  recall  the  geometric  construction  in  this  case  will  be  a 
help. 

Construct  Z  B  and  on  one  arm  lay  off  BC  =  a.  Then  with 
C  as  centre  and  h  as  radius  draw  a  circle  to  cut  BD.  If  b  <  a 
two  points  A  and  A'  will  be  found  (provided  h  is  long  enough 
to  reach  BD.  See  footnote,  p.  58).  If  6>a  only  one  point 
A  will  be  found. 

59.  Case  III.  Given  Two  Sides  and  the  Included  Angle. 
If  a,  h,  and  C  be  known  we  can  find  a  —  h,  a  -\-  h  and  ^{A  -\-  B) 
and  then  by  aid  of  a  formula  of  Art.  52,  l(A  —  B)  may  be 
calculated.  This  with  |(A  +  B)  will  give  A  and  B.  c  may 
then  be  found  by  the  law  of  sines. 


60  ELEMENTS  OF  PLANE  TRIGONOMETRY. 


Example.     Given  a 

=  42.38,  6  =  35  and  C 

=  43°  14'  40" 

solve  the  triangle. 

Formulae: 

tan      2       = 

a-b^     A+B 
=  a  +  6*^     2     ' 

asinC 
sm  A 

a  = 

42.38                      log  a  = 

11.62716  -  10 

b  = 

35               —  log  sin  A  = 

9.99565  -  10 

a-b  = 

7.38 

1.63151 

a  +  b  = 

77.38          +  log  sin  C  = 

9.83576  -  10 

A  +  B  =  1S0-C  = 

136°  45' 20"          logc  = 

1.46727 

UA  +B)  = 

68°  22' 40''            .-.  c  = 

29.327 

log  (a  —  6)  = 

0.86806 

+  log  tan  ^{A  +  B)  = 

0.40189 
11.26995  -  10 

-  log  (a  +  6)  = 

1.88863 

log  tan  i(A  -  5)  == 

9.38132  -  10 

.-.  i(A  -  B)  =  13°  31'  45" 
i(A  +B)  =  68°  22'  40" 
Adding,  A  =  81°  54'  25" 

Subtracting,  B  =  54°  50'  55" 

Hence  the  solution  is  A  =  81°  54'  25",  B  =  54°  50'  55"  and 
c  =  29.327. 

60.  Case  IV.  Given  Three  Sides.  If  a,  b,  and  c  are  given 
A/2,  J5/2  and  C/2  may  be  found  from  the  formulae  of  Art.  54. 
^  Example.  Given  a  =  0.312,  b  =  0.423  and  c  =  0.342,  find 
A,  B  and  C. 


Formulae : 

r           ^     B          r 

.      C         r 
tan^  - 

2       s  —  c 

a  =  0.312 

log  {s  -  a)  = 

9.35507  -  10 

b  =  0.423 

+  log  is  -b)  = 

9.06258  -  10 

c  =  0.342 

+  log  (s  -  c)  = 

9.29336  -  10 

2s  =  1.077 

27.71101  -  30 

s  =  0.5385 

-  log  s  = 

9.73119  -  10 

s  -  a  =  0.2265 

2117.97982  -  20 

s  -  b  =  0.1155 

I                         log  r  = 

8.98991  -  10 

s  -  c  =  0.1965 

s  =  0.5385 

ELEMENTS  OF  PLANE  TRIGONOMETRY. 


61 


log  r  =  18.98991  -  20 
-  log  (s-  a)  =    9.35507  -  10 
log  tan  A/2  =    9.63484  -  10 
.-.  A/2  =  23°  20' 
A  =  46°  40', 

log  r  =  18.98991  -  20 
log  {s  -h)  =    9.06258  -  10 
log  tan  B/2  =    9.92733  -  10 
.'.  5/2  =  40°  13'  45" 
B  =  80°  27' 30", 
log  r  =  18.98991  -  20 
log  (s-  c)  =    9.29336  -  10 
log  tan  (7/2  =    9.69655  -  10 
.-.  (7/2  =  26°  26'  15" 
C  =  52°  52'  30". 
61.  Example  1.     Each  of  two  ships  A  and   B,  415  yards 
apart,  measures  the  horizontal  angle  subtended  by  a  cliff  and 
the  other  ship;  the  angles  are  48°  17'  and  110°  10'  respectively. 
If  the  angle  of  elevation 
of  the  cliff  from  A  is  15° 
24'  what  is  the  height  of 
the  cliff? 

C  is  the  top  of  the  cliff 
and  CC  a  vertical  line,  C 
being  at  the  water  level. 
Then  it  is  given  that  AB 
=  415,  zC'AB  =  48°  17', 
ZC'BA  =  110°  10',  and 
ZCAC'  =  15°  24'.  It  is 
required  to  find  CC\ 

In  the  right  triangle 
ACC  we  can  find  CC  if  AC  be  known.     But  AC  may  be 
found  from  the  triangle  ABC^  since  three  parts  of  this  triangle 
are   given.     Use  the  law  of  sines  sin  B/AC  =  sin  C'/A 5  re- 
membering that  C  =  180  -  (A  +  B)  =  21°  33'. 
Solving, 

AB  sin  B      415  sin  110°  10' 


Fig.  43. 


AC  = 


sinC 


sin  21°  33' 


62 


ELEMENTS  OF  PLANE  TRIGONOMETRY. 


log  415  =  12.61805 

-  log  sin  21°  33'  =    9-56504 

3.0530T 

+  log  sin  110°  10'  =  9.97252 

.-.  log  AC  =  3.02553 

In  the  right  triangle  ACC 


10 
10 

10 


CC 


,  =  tan  15°  24'. 


AC 

.-.  CC  =  AC  tan  15°  24'. 
log  AC  =  3.02553 
+  log  tan  15°  24'  =  9.44004-10 
log  CC  =  2.46557 
.'.  CC  =  292.13. 

Example    2. 


Fig.  44. 


From  a 
tower  80  ft.  high,  objects 
A  and  B  in  a  plane  are 
found  to  have  angles  of 
depression  12°  52'  30"  and 
10°  41'  25"  respectively; 
the  horizontal  angle  at  C 
subtended  by  A  and  B  is 
43°  14'  40".  Find  the  dis- 
tance between  A  and  B. 
C  being  the  top  of  the 
tower  and  C  the  foot,  we 


have  given 

CC  =  80,  ZCAC  =  12°  52' 30". 
/  CBC  =  10°  41'  25"     and     zACB  =  43°  14'  40". 
In  the  right  triangles  ACC  and  BCC 

AC  =  80  cot  12°  52'  30",     BC  =  80  cot  10°  41'  25". 

log  80  =  1.90309 
+  log  cot  12°  52'  30"  =  0.64098 
log  AC  =  2.54407 
.-.  AC  =  350. 

log  80  =  1.90309 
+  log  cot  10°  41'  25"  =  0.72405 
log  BC  =  2.62714 
,'.BC  =  423.78. 


1. 

A  =  40°  20'  25'', 

2. 

A  =  52°  18'  25", 

3. 

A  =  57°  18', 

4. 

a  =  87.24, 

5. 

a  =  73  53, 

6. 

a  =  763.2, 

7. 

a  =  41  38, 

ELEMENTS  OF  PLANE  TRIGONOMETRY.  63 

Now  in  the  triangle  ABC  we  know  three  parts.  Hence  AB 
may  be  found  as  in  Art.  59. 

62.  Exercise  XIV.  Solve  the  triangles  in  which  the  following 
parts  are  given: 

B  =  75°  15'  35",     a  =  315.7. 

C  =  62°  18'  50",     h  =  42.72. 

a  =  3.715,  h  =  4.285 

h  =  73.58,  C  =  48°  17'. 

c  =  81.27,  B  =  72°  19'. 

h  =  653.5,  c  =  827.2. 

b  =  51.71,  c  =  47.82. 

8.  A  and  5  are  two  points  258  ft.  apart  and  C  a  point  such 
that  at  A  the  angle  subtended  by  BC  is  41°  18',  at  B  the  angle 
subtended  by  AC  is  72°  21'  5";  find  the  distance  between  A 
and  C. 

9.  A  and  B  are  points  400  ft.  apart  taken  on  the  edge  of  a 
river  and  C  is  a  stone  on  the  opposite  edge.  If  the  angles  CAB 
and  CBA  are  72°  20'  10"  and  65°  10'  40"  respectively  find  the 
width  of  the  river. 

10.  The  angle  of  elevation  of  an  object  from  the  foot  of  a 
hill  is  32°  15';  after  going  317  yds.  up  the  hill  away  from  the 
object  the  observer  finds  himself  on  a  level  with  it.  If  the 
slope  of  the  hill  is  15°,  find  the  distance  from  the  foot  of  the  hill 
to  the  object. 

11.  A  cliff  known  to  be  450  ft.  high  is  observed  to  be  due 
north  of  a  boat  and  at  an  elevation  of  30°.  After  going  a  dis- 
tance northeast  the  boat  found  the  elevation  to  be  35°.  How 
far  did  it  go? 

12.  From  a  point  on  one  side  of  a  muskeg  a  man  measures 
450  yds.  east;  then  northwest  a  distance  of  652  yds.  How  far 
is  he  from  the  starting  place? 

13.  The  sides  AB,  BC  and  CA  of  a  triangle  are  250,  300 
and  350  ft.  respectively.  BC  is  produced  275  ft.  to  the  point  D. 
What  is  the  angle  at  A  subtended  by  CDl 

14.  At  each  of  two  places  400  ft.  apart  the  elevation  of  a  kite 
is  found  to  be  27°  15'.  The  horizontal  angle  at  one  place 
subtended  by  the  kite  and  the  other  is  50°  20'.  Find  the  height 
of  the  kite. 


64  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

15.  At  noon  the  shadow  of  a  cloud  falls  450  ft.  to  the  north 
of  an  observer;  if  the  elevation  of  the  cloud  (which  is  south 
of  the  observer)  be  75°  12'  and  of  the  sun  be  62°  8'  what  is  the 
vertical  height  of  the  cloud? 

16.  In  15  what  would  be  the  height  were  the  cloud  north  of 
the  observer? 

17.  A  lighthouse  is  75  ft.  high  and  has  an  elevation  25°  19' 
from  one  point;  from  another  point  120  ft.  from  the  first  the 
elevation  is  28°  23'  40".  What  is  the  horizontal  angle  at  the 
lighthouse  subtended  by  the  two  points? 

18.  The  elevation  of  a  balloon  due  north  of  a  station  is  30°; 
at  a  place  one  mile  southeast  of  the  first  the  horizontal  angle 
subtended  by  the  balloon  and  the  first  place  is  20°  15'.  Find 
the  height  of  the  balloon. 

19.  The  elevation  of  a  rock  is  47°;  after  walking  1,000  ft. 
toward  it  up  a  hill  inclined  at  32°  to  the  level  the  elevation 
is  77°.     Find  the  height  of  the  rock  above  the  first  point. 

20.  From  the  foot  of  a  tree  in  a  level  field  a  line  200  ft.  is 
measured  and  the  elevation  of  the  top  of  the  tree  is  found  to 
be  18°  20'.  From  this  point  a  line  325  ft.  long  is  measured 
and  the  elevation  of  the  top  is  then  found  to  be  22°  30'  25". 
Find  the  angle  between  the  two  lines  which  were  measured. 

21.  A  triangular  piece  of  ground  is  found  to  be  37  ft.  6  in. 
by  42  ft.  3  in.  by  58  ft.  3  in.  Find  the  angle  opposite  the 
shortest  side. 

22.  From  a  tower  75  ft.  high  the  angles  of  depression  of  two 
objects  are  24°  29'  and  31°  58';  the  horizontal  angle  subtended 
by  the  objects  is  51°  42'.  Find  the  distance  between  the 
objects  (supposed  to  be  level  with  the  foot  of  the  tower). 

23.  From  a  mountain  top  3,200  ft.  above  sea  level  ships 
are  observed,  one  east,  the  other  southeast;  the  angles  of  de- 
pression are  12°  43'  and  15°  37'.  Find  the  distance  between 
the  ships,  and  the  direction  from  one  ship  to  the  other. 

24.  Two  towns  A  and  B  are  four  miles  apart ;  from  a  balloon 
above  A  the  depression  of  B  is  15°  6'  and  when  the  balloon  is 
above  B  the  depression  of  A  is  19°  30'.  Find  how  much  the 
balloon  has  risen. 

25.  The  horizontal  distance  between  two  points  A  and  B 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  65 

is  a  mile  and  a  half.  The  horizontal  angle  at  A  subtended  by 
B  and  the  top  of  a  mountain  is  65°  18'  25"  and  that  at  B  sub- 
tended by  A  and  the  top  is  74°  29'  40''.  The  angle  of  elevation 
at  A  of  the  top  of  the  mountain  is  18°  28'  55".  If  the  height 
above  sea  level  of  A  is  2,500  ft.  find  the  height  of  the  mountain. 

26.  A  flagstaff  78  ft.  high  stands  on  the  face  of  a  hill  whose 
inclination  to  the  horizon  is  34°.  At  a  point  down  the  hill 
from  the  flagstaff,  the  angle  of  elevation  of  its  top  is  57°  28'; 
find  the  distance  from  the  observer  to  the  foot  of  the  flagstaff. 

27.  At  a  point  up  the  hill  from  the  flagstaff  in  (26),  the  de- 
pression of  its  top  is  18°  53'.  Find  the  distance  of  the  observer 
from  the  top  of  the  flagstaff. 

28.  Two  stations  A  and  5  on  a  level  plane  are  785.4  ft.  apart. 
At  A  the  elevation  of  an  aeroplane  C  is  38°  19'  25",  and  the 
horizontal  angle  at  A  subtended  by  B  and  C  is  41°  30'  20". 
At  B  the  horizontal  angle  subtended  by  A  and  C  is  63°  18'  20". 
Find  the  height  of  the  aeroplane. 

29.  The  observer  at  B  in  (28)  makes  the  elevation  of  C 
46°  49'  at  the  same  time  that  the  other  observations  were  taken. 
Find  the  height  of  C  using  this  observation. 


CHAPTER  V. 


SPHERICAL  TRIGONOMETRY. 

Introductory  Review. 

63.  Those  definitions  and  theorems  of  sohd  geometry  which 
are  essential  to  the  study  of  spherical  trigonometry  will  be  stated 
or  briefly  discussed  in  this  chapter. 

64.  Diedral  Angles.  Let  ABC  and  ABD  be  two  planes 
intersecting  in  the  line  AB.  The  figure  formed  by  these  planes 
at  their  intersection  is  called  a  diedral  angle.  The  line  AB  is 
called  the  edge  of  the  diedral  angle.  Let  P  be  any  point  on 
the  edge,  and  let  PE  and  PF  be  lines  drawn  perpendicular  to 
AS  in  the  planes  ABC  and  ABD  respectively.     The  angle  EPF 


Fig.  45. 


Fig.  46. 


is  called  the  plane  angle  of  the  diedral  angle,  and  is  the  measure 
of  the  diedral  angle. 

65.  Triedral  Angles.  Let  ABC,  ACD,  and  ADB  be  three 
planes  meeting  in  the  common  point  A.  The  figure  formed 
by  these  planes  is  called  a  triedral  angle.  The  point  A  is 
called  the  vertex;  the  intersections  of  the  planes  AB,  AC,  AD 
are  called  the  edges;  the  portions  of  the  planes  between  the 
edges  are  called  the  faces;  the  angles  formed  at  the  vertex 
by  the  edges  BAC,  CAD,  DAB  are  called  the  face  angles;  and 
the  diedral  angles  formed  at  the  edges  are  called  the  diedral 
angles  of  the  triedral  angle. 

66.  The  Sphere.     A  spherical  surface  is  a  surface  all  points 

66 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  67 

of  which  are  equidistant  from  a  point  called  the  centre.  A 
sphere  is  a  solid  bounded  by  a  spherical  surface. 

Every  plane  section  of  a  sphere  is  a  circle.  A  great  circle  on 
a  sphere  is  a  circle  whose  plane  passes  through  the  centre  of 
the  sphere.     All  other  circles  on  a  sphere  are  called  small  circles. 

Let  A  and  B  be  two  points  on  the  surface  of  a  sphere,  and 
let  AB  he  the  shorter  arc  of  the  great  circle  passing  through 
these  points.  Then  AB  is  the  shortest  path  that  can  be  drawn 
on  the  surface  of  the  sphere  between  A  and  B.  The  length  of 
this  arc  is  defined  as  the  distance  from  A  to  B  measured  on 
the  surface  of  the  sphere.  Let  0  be  the  centre  of  the  sphere, 
and  draw  OA  and  OB.  Then, 
since  on  the  same  or  equal  circles 
equal  arcs  subtend  equal  angles 
at  the  centre,  the  angle  AOB 
may  be  tsken  as  the  measure  of 
the  arc  AB.  The  angle  AOB  is 
called  the  angular  distance  be- 
tween A  and  B.  Produce  BO 
to  meet  the  sphere  again  in  Bi. 

Let  the  radius  of  the  sphere  be  -^      ^» 

^  Fig.  47. 

r,  and  let  the  angle  AOB  (meas- 
ured in  degrees)  be  6°.     Then,  from  plane  geometry,  we  have 

arc  AB  :  arc   BBi  =  0  :  180 
But  arc  BBi  =  a  semicircle  =  irr. 

Therefore 

^'•^  ^^ = iS- 

Thus,  given  the  angular  distance  6°  and  the  radius  of  the  sphere^ 
the  actual  distance  on  the  surface  of  the  sphere  may  be  com- 
puted. 

Two  points  are  said  to  be  at  a  quadrant's  distance  when  their 
angular  distance  is  90° 

The  poles  of  any  circle  on  a  sphere  are  the  extremities  of 
the  diameter  of  the  sphere  drawn  perpendicular  to  the  plane 
of  the  circle.  Each  pole  of  a  great  circle  is  at  a  quadrant's 
distance  from  every  point  of  the  circle 

67.  Spherical  Angles.     The  angle  between  two  circles  on  the 


68 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 


Fig.  48. 


surface  of  a  sphere  is  defined  as  the  angle  between  the  tangents 

to  the  circles  at  their  point  of  intersection.     The  angle  formed 

by  the  arcs  of  two  great  circles  is  called  a  spherical  angle. 
Let  AB  and  AC  be  arcs  of  two  great  circles  on  a  sphere  whose 

centre  is  at  0,  and  let  AS  and  AT  be  the  tangents  to  these 

circles  at  A. 

Then  by  definition  the  spherical  angle  is  measured  by  the 

angle  TAS.     But  since  a  tangent  to  a  circle  is  perpendicular 

to  the  radius  drawn  to  the  point  of 
contact,  AT  and  AS  are  perpen- 
dicular to  OA,  and  hence  TAS  is 
the  measure  of  the  diedral  angle 
T-OA-S  or  B-OA-C,  A  spheri- 
cal angle  is  therejore  measured  hy 
the  plane  angle  of  the  diedral  angle 
formed  hy  the  planes  of  the  circles. 
Draw  OB  and  OC  perpendicular 
to  OA  at  0  in  the  planes  AOB 
and  AOC  respectively.     Then  the 

angles  TAS  and  BOC  are  equal.     But  BOC  is  measured  by  the 

great  circle  arc  BC.     Hence,  since  A  is  the  pole  of  the  great 

circle  arc  BC,  a  spherical  angle  is  measured  hy  the  arc  of  the  great 

circle  of  which  the  vertex  is  a  pole,  and  which  is  intercepted  hetween 

the  sides  of  the  angle. 
68.  Spherical  Triangles.    The 

figure  bounded  by  the   arcs   of 

three    great    circles   is    called   a 

spherical  triangle.     Let  ABC  be  a 

spherical    triangle    on   a   sphere 

whose  centre  is  at  0.    Draw  the 

radii  OA,  OB  and  OC.    Then  the 

figure  formed  by  the  planes  OAB, 

OAC,    and    OBC    is    a    triedral 

angle.    The  angles  of  the  spherical 

triangle    are    measured    by    the 

plane  angles  of  the  diedral  angles  of  this  triedral  angle.     Also 

the  sides  of   the  spherical  triangle  are  measured  by  the  face 

angles  of  the  triedral  angles.     For,  by  definition  the  angular 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  69 

distance  AB  is  the  angle  A 05,  subtended  by  AB  at  the  centre. 
Hence  with  any  spherical  triangle  there  is  associated  a  triedral 
angle  at  the  centre  of  the  sphere,  such  that  the  angles  of  the 
triangle  are  measured  by  the  diedral  angles  of  the  triedral 
angle,  and  the  sides  of  the  triangle  by  the  face  angles  of  the 
triedral  angle.  This  complete  correspondence  between  the 
spherical  triangle  and  the  triedral  angle  at  the  centre  of  the 
sphere  is  of  fundamental  importance  in  what  follows. 

69.  It  is  usual  to  assume  that  each  side  of  a  spherical  triangle 
is  less  than  180°.     This  assumption  causes  no  loss  of  generality. 
For  let  AB{D)C  he  SL  spherical  triangle 
in  which  the  side  BC  is  greater  than 
180°.  Complete  the  great  circle  BDCE. 
Then  in  the  triangle  AB{E)C  each  of 

the  sides  Is  less  than  180°.     We  may 

then  consider  this  triangle  instead  of 

the  original  one,  and  at  the  end  of  our 

investigations  we  may  return  to  the 

triangle   AB{D)C  by  taking  the  sup- 
plements of  the  angles  B   and  C  of 

AB{E)C,  for  the  corresponding  angles 

oiAB(D)C,  and  by  takins  for  BDC  the  difference  between  360° 

and  BEC. 

It  is  easily  shown  (by  producing  BA  to  meet  BDC  in  D) 

that  the  angle  A  is  greater  than  180°  if  the  side  opposite  to  it, 

BDC,  is  greater  than  180°,  and  conversely. 

The  sum  of  any  two  sides  of  a  spherical  triangle  is  greater  than 

the  third  side.     This  follows  from  the  corresponding  theorem 

concerning  the  face  angles  of  a  convex  triedral  angle. 

The  sum  of  the  sides  of  a  spherical  triangle  is  less  than  four 

right  angles,  since  the  sum  of  the  face  angles  of  a  convex  triedral 

angle  is  less  than  four  right  angles. 
70.  Polar  Triangles.     Let  arcs  of  great  circles  be  drawn  with 

the  vertices  of  a  spherical  triangle  ABC  as  poles,  and  let  C 

be  that  intersection  of  the  circles  drawn  with  A  and  B  as  poles 

which  lies  on  the  same  side  of  the  arc  A 5  as  does  C,  and  similarly 

for  the  other  two  intersections  B'  and  A'.     The  triangle  A'B'C 

is  said  to  be  the  polar  triangle  of  ABC, 


70 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 


If  the  first  of  two  spherical  triangles  is  the  polar  of  the  second, 
then  the  second  is  also  the  polar  of  the  first. 

In  two  polar  triangles  each  angle  of  one  is  the  supplement  of 
the  side  opposite  to  it  in  the  other. 

The  latter  statement  may  be  proved  as  follows. 
Let  ABC  and  A'B'C  be  two  polar  triangles. 
To  prove  that  A  =  180°  -  a\  etc. 

Produce  the  arcs  AB  and  AC 
to  meet  B'C  in  M  and  N  re- 
spectively. Since  B'  is  the  pole 
of  the  arc  ACN,  B'N  is  a  qua- 
drant. Similarly  MC  is  a  qua- 
drant. Hence  B'N  +  MC  = 
180°.  That  is,  B'N  +  NC  + 
MN  =  180°.  But  B'N  +  N'C 
=  B'C  =  a',  and  MN  is  the 
measure  of  the  angle  A,  (page 
""•  68).     Therefore  a'  +  A  =  180°, 

or  A  =  180°  -  a',  etc. 

71.  The  sum  of  the  angles  of  a  spherical  triangle  is  greater 
than  two  and  less  than  six  right  angles. 
Let  ABC  be  a  spherical  triangle. 
To  prove  that  180°  <A+B  +  C<  540°. 
Construct  the  polar  triangle.     Then, 

A  =  180°  -a',    B  =  180°  -h',     C  =  180°  -  c'. 

Hence,  upon  addition, 

A  +  B  -{-C  =  540°  -  {a'  +  h'  +  c'). 
Now 

0°  <  a'  +  6'  +  c'  <  360°. 
Hence 

A  -{-  B  +  C  =  540°  -  [something  less  than  360°], 

that  is, 

A  +  B-\-C>  180°. 
Similarly, 

A  -\-  B  +  C  =  540°  -  [something  greater  than  0°], 

that  is, 

A  +  B  +  C  <  540°. 
Therefore 

180°  <A  +  B  +  C<  540°. 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  71 

The  spherical  excess  of  a  triangle  is  defined  as  the  difference 
between  the  sum  of  the  angles  of  the  triangle  and  180°. 

72.  Lunes.  The  figure  formed  by  joining  the  extremities  of  a 
diameter  of  a  sphere  by  the  arcs  of  two  great  circles  is  called  a 
lune.  The  extremities  of  the  diameter  are  called  the  vertices 
of  the  lune,  and  the  spherical  angles  formed  at  the  vertices  by 
the  great  circle  arcs  are  called  the  angles  of  the  lune.  Thus 
in  Fig.  50  BADCB  is  a  lune  with  vertices  at  B  and  Di. 

73.  Spherical  Degrees.  Let  A  and  B  be  the  ends  of  a 
diameter  of  a  sphere.  With  A  and  B  as  vertices  construct  a 
lune  whose  angle  is  1°.  Then  the  area  of  this  lune  is  one  three 
hundred  and  sixtieth  of  the  area  of  the  sphere.  If  now  we 
draw  with  A  and  B  as  poles  the  great  circle  arc  CD,  the  lune 
ACBD  is  divided  into  two  equal  triangles.  The  area  of  one 
of  these  triangles  is  defined  as  a  spherical  degree  of  area,  or  as  a 
spherical  degree.  The  total  surface  of  the  sphere  then  contains 
720  spherical  degrees. 

Since  the  areas  of  two  lunes  on  the  same  or  equal  spheres 
are  proportional  to  their  angles,  a  lune  of  angle  A°  contains  2 A 
spherical  degrees. 

74.  Areas  of  Spherical  Triangles.  We  shall  now  show  that 
the  area  of  a  spherical  triangle,  measured  in  spherical  degrees,  is 
equal  to  the  spherical  excess  of  that  triangle. 

Let  ABC  be  a  spherical  triangle.     Complete  the  great  circle 
arcs  AB,  BC,  CA,  the  two  circles 
through  A  meeting  again  in  A], 
etc. 

The  arcs  ACi  and  AiC  are 
equal  since  each  is  equal  to  a 
semicircle  diminished  by  the  arc 
AC.  Similarly,  BC  =  BiCi,  and 
ABi  =  AiB.  Then  the  two  tri- 
angles AiBC  and  ABiCi  are 
equivalent. 

Hence,  the  area  of  the  June 
ABAiC  =  ABC  +  AB^Ci  =  2A  spherical  degrees. 

Also,  the  area  of  the  lune  BAB^C  =  ABC  +  ABiC  =  2B 
spherical  degrees. 


72  ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 

And  the  area  of  the  lune  CACiB  =  ABC  +  ACiB  =  2C 
spherical  degrees.     Hence,  on  addition, 

2ABC  +  the  hemisphere  CBCiBi  =  2{A -\-  B -\- C) 
or 

the  area  of  ABC  =  A  -\-  B  -\-  C  —  ISO  spherical  degrees. 

Hence  the  area  of  a  spherical  triangle  is  measured  by  its  spherical 
excess. 

The  total  area  of  a  sphere  of  radius  r  is  4:Tr^.  Also  there  are 
720  spherical  degrees  in  the  surface  of  a  sphere.  Hence  the 
area  of  one  spherical  degree  is  equal  to  47rrV720  or  TrrVlSO. 
It  follows  then  that  the  area  of  a  spherical  triangle  is  given  by 
the  formula 

.  P^      {A+B  +  C  -  180)7rr2 
area  ABC  =  ^ ^^ ^—. 


CHAPTER  VI. 

RELATIONS   BETWEEN   THE  SIDES   AND   ANGLES   OF  A 
SPHERICAL  TRIANGLE. 

75.  The  Law  of  Cosines.  Let  ABC  bo  a  spherical  triansle 
on  a  sphere  wliose  centre  is  at  0,  and  let  the  sides  6  and  c  bo 
acute. 

At  any  point,  Ai,  on  OA  draw  a 
plane  perpendicular  to  OA  meeting 
the  planes  OAC,  OAB,  and  OBC  in 
AiL,  AiM,  and  ML  respectively. 

Then  MAJj  is  the  measure  of  the 
diedral  angle  B-OA-C,  and  therefore 
measures  the  spherical  angle  A  [Art.  Fig.  63. 

G7].     Also  the  angles  OAiM  and  OAJj  are  right  angles  by  con- 
struction. 

In  the  triangle  MAiLy 

MU-  =  MAi^  +  LAi^  -  2MAvLA,  cos  A, 
and  in  the  triangle  MOL, 

MU  =  M02  +  W  -  2M0-L0  cos  a. 
Hence,  upon  subtraction  and  transposition, 

2M0L0  cos  a  =  MC  -  MAt" 

+  LO^  -  LAi'  +  2MAi'LAr  cos  A.    ^^^ 
But,  since  the  triangles  MOAi  and  LOAi  are  right  triangles, 
MO'  -  MAi"  =  OAi"    and    LO'  -  LAi"  =  OAi", 

The  equation  (1)  may  then  be  written 

MO'LO  cos  a  =  OAi^  -{-  MAi-LArcon  A, 
or 

OAi  OAx  ,  MAx  LAi 


But 


''''''==  MO'lJ"^  MO' LO'''''^' 

OAl  Tirr^A  MA\  .        Tijrr\A 

v^^  =  cos  MOAi  =  cos  c,      ^.  =  sin  MOAi  =  sm  c, 

-y—  =  cos  LOAi  =  cos  6,      Y^  =  sin  LOAi  =  sin  h. 
Li)  IjU 

73 


74  ELEMENTS   OF   SPHERICAL  TRIGONOMETRY. 

Hence  cos  a  =  cos  h  cos  c  +  sin  6  sin  c  cos  A.  (2) 

It  was  necessary  to  assume  that  the  sides  h  and  c  were  acute 

in  order  to  draw  the  right  triangles  LOAi  and  MOAi  as  in  the 

figure.     We  shall  now  show  that  the  formula  (2)  is  true  for 

any  spherical  triangle. 

Case  (1).     Let  ABC  be  a  spherical  triangle  in  which  h  and 

c  are  both  obtuse. 

Produce  the  arcs  AB  and  AC  to  form  the  lune  AA\     Then 

in  the  triangle  AiBC,  hi  and  ci  are  acute  and  equal  to  180°  —  h 


and  180°  —  c  respectively.     Then  the  formula  (2)  is  true  for 
this  triangle.     That  is, 

cos  a  =  cos  (180°  -  b)  cos  (180°  -  c) 

+  sin  (180°  -  h)  sin  (180°  -  c)  cos  A 
or 

cos  a  =  cos  h  cos  c  +  sin  h  sin  c  cos  A.     • 

Case  (2).  Let  ABC  be  a  spherical  triangle  in  which  h  is 
acute  and  c  obtuse.  Produce  the  arcs  BA  and  BC  to  form  the 
lune  BBi.     Then  in  the  triangle  ABiC,  b  and  Ci  are  acute. 

Hence, 

cos  tti  =  cos  b  cos  Ci  +  sin  b  sin  Ci  cos  CABi. 
But 

ai  =  180°  -  a,  ci  =  180°  -  c. 
and 

CABi  =  180°  -  CAB  =  180°  -  A. 

Therefore 

cos  a  =  cos  b  cos  c  +  sin  b  sin  c  cos  A. 

Hence  in  any  spherical  triangle 

cos  a  =  cos  b  cos  c  +  sin  6  sin  c  cos  Ay 
and  similarly  cos  6  =  cos  c  cos  a  +  sin  c  sin  a  cos  By  I 

and  cos  c  =  cos  a  cos  fe  +  sin  a  sin  6  cos  C 

76.  Since  the  equations  (I)  are  true  for  any  triangle,  they 
are  true  for  the  polar  triangle  A'B'C  of  a  given  triangle  ABC. 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.  75 

Hence, 

cos  a'  =  cos  h'  cos  c'  +  sin  h'  sin  c'  cos  A'. 
But 

a'  =  180°  -  A,  6'  =  180°  -  B,  c'  =  180°  -  C. 
and 

A'  =  180°  -  a. 
Therefore, 

—  cos  A  =  (—  cos  B){—  cos  C)  +  sin  B  sin  C(—  cos  a) 
or 

cos  i4  =  —  cos  B  cos  C  +  sin  B  sin  C  cos  a, 

similarly,  cos  B  =  —  cos  C  cos  A  +  sin  C  sin  A  cos  6,  II 

and  cos  C  =  —  cos  A  cos  5  +  sin  A  sin  B  cos  c. 

77.  The  Sine  Proportion.     From  the  first  equation  of  (I), 

Art.  75,  we  have 

.       cos  a  —  cos  b  cos  c 

cos  A  = . — ^ — ^ : 

sm  0  sm  c 

Hence 

(cos  a  —  cos  h  cos  c)^ 


1  -  cos^A  =  1 


sin^  h  sin^  c 
or 

.  „   J,       sin^  6  sin^  c  —  (cos  a  —  cos  h  cos  c) 


sin^  6  sin^  c 


_  (1  —  cos^  6)(1  —  cos^  c)  —  (cos  a  —  cos  h  cos  c)^ 
sin^  b  sin^  c 
Therefore, 


sin^  A  _1  —  cos^  a  —  cos^  b  —  cos^  c  +  2  cos  a  cos  6  cos  c 

sin^  a  sin^  a  sin^  b  sin^  c 

or  

sin  A  _  l/l  —  cos^  a  —  cos^  6  —  cos^  c  -\-  2  cos  a  cos  b  cos  c 
sin  a  sin  a  sin  6  sin  c  ' 

the  positive  sign  being  taken  since  A  and  a  are  each  less  than 
180°. 

The  expression  on  the  right  is  symmetric  in  the  letters  a,  b, 
and  c.  That  is,  if  any  two  of  the  letters  be  interchanged,  the 
expression  is  unaltered.  Then,  if  we  proceed  as  above  with  the 
second  and  third  equations  of  (I),  page  74,  we  shall  obtain  on 
the  left  sin  5/sin  b,  and  sin  C/sin  c,  and  on  the  right  the  same 
expression  as  above.     Hence, 

sin  A      sin  B      sin  C 


sin  a      sin  &      sin  c 


m 


76  ELEMENTS   OF   SPHERICAL   TRIGONOMETRY. 

78.  Law  of  Tangents.     From  Art.  42,  page  43,  we  have 


.A           1  —  cos  A  .^. 

tan  -  =  ^  -— J,  (1) 

and  from  (I),  page  74, 

.       cos  a  —  cos  h  cos  c  ,^. 

cos  A  = r— r-^ ■.  (2) 

sm  6  sm  c  ^  ^ 

Substituting  in  (1)  the  value  of  cos  A  found  from  (2),  we  obtain 


cos  a  —  cos  6  cos  c 


tan  4  =      '  ^™  **  ^i°  *= 


2  / 1    I   ^<^s  ^  "~  ^^^  ^  ^^^  ^ 


sin  b  sin  c 


oin  6  sin  c  —  cos  a  +  cos  b  cos  c 
sin  6  sin  c  +  cos  a  —  cos  b  cos  c 

cos  6  cos  c  +  sin  6  sin  c  —  cos  a 
cos  a  —  (cos  6  cos  c  —  sin  6  sin  c) 


cos  (b  —  c)  —  cos  g 
cos  a  —  cos  (6  +  c) 

_      /  cos  a  —  cos  (6  —  c) 
\  cos  (6  +  c)  —  cos  a 
Or,  upon  making  use  of  a  factor  formula  (page  44), 


tan  2  = 

Isin  -i(a  +  6  -  c)  sin  ■^(a  -  6  +  c) 

\sin  i(6  +  c  +  a)  sin  i(6  +  c  -  a)' 

Let 

a  +  6  +  c  =  2s. 

Then 

-  a  +  6  +  c  =  2s  -  2a, 

and  so 

•i(-  a  +  6  +  c)=s-a, 

and 

■i(a  —  6  +  c)  =  s  —  6, 

and 

1  (a  +  6  -  c)  =  s  -  c. 

Then 

A 
2 

tan 

sin 

(s  —  b)  sin  (s  —  c) 
sin  s  sin  (s  —  a) 

1  jsi 

sin  (s  —  a)    \ 


sin  (s  —  g)  sin  (s  —  b)  sin  (s  —  c) 
sin  s 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  77 

Now  let 

j^  _      /sin  (s  —  a)  sin  (s  —  b)  sin  (s  —  c)* 


"~  A 

J                         sin  s 

We  then  have, 

observing  that  K  is  symmetric  in  a,  h,  c, 

.     A             K 

2     sin  (s  -  a) 

similarly 

B            K 

2      sin  (s  —  o) 

and 

.    c        a: 

tan  _  =  _,  -,.      _,. 

IV 


2      sin  (s  —  c) 
79.  Napier's  Analogies.     From  the  equations  above  we  have 

tan  —  tan  —  = 


2  2      sin  (s  —  a)  sin  (s  —  6)* 

Or,  upon  replacing  each  tangent  by  the  ratio  of  the  sine  to  the 
cosine  and  substituting  for  K^  its  value, 


.    A   .    B 

sm  — sm^,       .    r         V 
2_ 2  _  sm  (s  —  c) 

A       B  sins 

cos -cos  2 


(1) 


Subtracting  each  side  from  unity  and  simplifying, 

A       B        .    A   .    B 

cos  —  COS  -  -  sm  —  sm  -        .  .    .         . 

2         2  2         2  _  sm  s  —  sm  [s  —  c) 

A        B  sins  * 

cos  -  COS  2 

Making  use  of  the  formula  for  the  cosine  of  the  sum  of  two 
angles,  page  38,  and  that  for  the  difference  between  the  sines  of 
two  angles,  page  44,  the  equation  above  becomes 


1  /  i    I    TDN      2  cos  4(2s  —  c)  sin 
cos|(A  +B)  _  2^  ^ 


c 


A       B  sins  •  ® 

cos  —  COS  -^ 

If  now  we  add  unity  to  each  side  of  (1)  and  simplify  the  result 
in  a  manner  precisely  similar  to  the  above,  we  shall  obtain 
*  For  the  geometrical  interpretation  of  K  see  page  101. 


78  ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 

/» 

1  /  ^       r>N      2  sin  U2s  —  c)  cos  - 
cos \{A  —  B)  _  ^^  ^       2 

A       B  sin  s 

cos  2  cos  - 

Upon  dividing  (2)  by  (3)  and  noting  that 

2s  —  c=  (a  +  6  +  c)  —  c  =  a-{-h, 
we  have 

cosK^  +  B)  ^  ''°^K«  +  &)sin| 
cosKA-B)      ,inKa  +  6)cos| 

=  cot  |(a  +  6)  tan-. 


(3) 


Therefore 


cosjOM-BJ  _  _J^ 


V(a) 


cos-i(i4-5)      tan-i(a  +  6)* 
From  (IV)  we  also  have 

2  _  sin  (s  —  6) 

B      sin(s  — a)' 
tan  2       • 

By  replacing  each  tangent  by  the  ratio  of  the  sine  to  the  cosine, 

this  equation  becomes 

.   A        B 

sm-  COS-pr  ,     f         ,x 

2        2  _  sm  (s  —  o)  ... 

A   .""B"sin(s-a)*  ^  ^ 

cos  ^  sm  - 

If  now  we  apply  to  (4)  the  same  series  of  operations  that  we 
have  above  applied  to  (1),  we  obtain 

sini(A-B)    tani(a-6)*  ^"^^ 

The  details  of  this  reduction  are  left  for  the  student. 

Since  V(a)  is  true  for  any  spherical  triangle,  it  is  true  for 
the  polar  triangle  A'B'C  of  a  given  triangle  ABC.     That  is, 

cosKA^  +  gQ  ^         ^^""2  .. 

cos  \{A'  -  B')      tan  \{a'  +  6')' 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.  79 

But, 

i(A'  +  50  =  i(180°  -  a  +  180°  -h)  =  180°  -  i(a  +  6), 
■|(A'  -  B')  =  i(180°  -  a  -  180°  +  6)  =  -  ^(a  -  5), 
i(a'  +  6')  =  i(180°  -  A  +  180°  -h)  =  180°  -  -i(A  +  B), 

and 

1^  =  4(180°- C)  =  90°- |. 

Substituting  these  values  in  (5)  we  have 

cos4(a-6)      tani(A  +  5)'  ^^ 

Similarly,  by  aid  of  the  polar  triangle,  V(6)  gives  rise  tc 

•    1  /     .  L\  cot  - 

sini(a  +  6)_  2 ^  . 

sini(a-6)    taniiA-BY  ^^""^ 

The  four  equations  found  in  this  article  are  known  as  Napier's 
analogies.  Each  of  these  equations,  upon  permuting  the  letters, 
gives  rise  to  two  new  equations.  In  all,  then,  there  are  twelve 
such  equations  of  which  a,  h,  c  and  d  are  the  types. 

80.  On  Species.  Two  angles  are  said  to  be  of  the  same 
species  if  they  are  both  acute  or  both  obtuse,  and  of  different 
species  if  one  is  acute  and  the  other  obtuse. 

The  following  law  of  species  is  of  great  importance  and  should 
be  carefully  memorized. 

One  half  the  sum  of  any  two  sides  of  a  spherical  triangle  and 
one  half  the  sum  of  the  two  opposite  angles  are  of  the  same  species. 
For,  from  V(a), 

cosUA  +  B)  ^       ^^"^2 
cos  ^{A  —  B)     tan  ^(a  +  b)' 

Since  each  side  and  angle  of  the  spherical  triangle  is  less  than 
180°  [Art.  69,  page  69],  i(A  +  B)  and  4(a  +  b)  are  each  less 
than  180°,  and  ^{A  —  B)  and  c/2  are  each  less  than  90°.  Hence, 
in  the  above  equation,  the  denominator  on  the  left  and  the 
numerator  on  the  right  are  each  positive.  Then  cos  \{A  +  B) 
and  tan  -|(a  +  6)  are  either  both  positive  or  both  negative,  and 
therefore  \{A  -\-  B)  and  \{a  -\- b)  are  of  the  same  species  . 


CHAPTER  VII. 


THE  RIGHT  SPHERICAL  TRIANGLE. 


81.  A  spherical  triangle  one  of  whose  angles  is  a  right  angle 
is  called  a  right  spherical  triangle. 

We  shall  prove  [Art.  85]  that  if,  in  addition  to  the  right  angle, 
two  other  parts  be  given  the  triangle  can  be  solved.  For  this 
purpose  we  deduce  from  formulae  of  the  preceding  chapter 
ten  relations  each  involving  a  different  set  of  three  parts.  These 
formulae  can  be  conveniently  recalled  by  means  of  Napier^s 
rule  of  circular  parts. 

By  the  circular  parts  of  a  right  spherical  triangle  we  mean 
the  two  sides  about  the  right  angle  and  the  complements  of 
the  hypotenuse  and  the  other  two  angles.     Place  these  five 

parts  as  in  the  accompanying 
figure,  being  careful  to  arrange 
them  in  the  order  in  which  they 
occur  in  the  triangle.  It  will  be 
noticed  that  on  this  figure  any 
three  parts  can  be  classed  as 
either  a  middle  and  two  adjacent 
parts,  or  a  middle  and  two  op- 
posite parts. 

82.  Napier's  Rule.  The  sine  of  the  middle  part  is  equal  to 
the  product  of  the  tangents  of  the  adjacent  parts,  and  to  the  product 
of  the  cosines  of  the  opposite  parts. 

Since  there  are  ten  combinations  of  five  things  taken  three 
at  a  time,  this  rule  will  give  us  ten  formulae.  For  example, 
let  ABC  be  a  right  spherical  triangle  right-angled  at  C.  Then 
taking  b,  co-A,  co-B  the  middle  part  is  co-B  and  the  other  two 
are  the  opposite  parts.     Therefore, 


Fig.  56. 


sin  (90°  -  B)  =  cos  (90°  -  A)  cos  6, 


or 


cos  B 


sin  A  cos  b. 
80 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  81 

Also  taking  a,  co-c,  co-B,  the  middle  part  is  co-B  and  the  others 
are  the  adjacent  parts.     Therefore, 

sin  (90°  -  B)  =  tan  a  tan  (90°  -  c), 
or 

cos  B  =  tan  a  cot  c 

The  following  table  contains  all  the  formulae  so  obtained : 

cos  c  =  cos  a  cos  6,     (1)        cos  c  =  cot  A  cot  B,      (2) 

cos  A  =  sin  B  cos  a,     (3)       cos  B  =  sin  A  cos  h,       (4) 

sin  a  =  sin  c  sin  A,      (5)       sin  6  =  sin  c  sin  B,         (6) 

sin  a  =  tan  6  cot  Bj     (7)       sin  6  =  tan  a  cot  A,       (8) 

cos  A  =  tan  h  cot  c,      (9)      cos  5  =  tan  a  cot  c.      (10) 

83.  These  relations  may  be  deduced  as  follows: 
Upon  setting  C  =  90°,  the  last  equation  of  (I),  page  74, 
becomes 

cos  c  =  cos  a  cos  b.  (1) 

The  last  equation  of  (II),  page  75,  gives 

0  =  —  cos  A  cos  5  +  sin  A  sin  B  cos  c, 

and  therefore, 

cos  c  =  cot  A  cot  ^.  (2) 

The  first  two  equations  of  (II),  page  75,  give 

cos  A    =  sin  5  cos  a,  (3) 

and 

cos  B  =  sin  A  cos  h  (4) 

The  sine  proportion  (III),  Art.  77,  becomes 

sin  A      sin  B         1 


and  hence 
and 


sin  a 

sin  b       sin  c 

sin  a 

=  sin  c  sin  A, 

sin  b 

=  sin  c  sin  B, 

sin  ( 

sin  6  sin  A 

(5) 
(6) 


sin  B 
Or,  upon  substituting  for  sin  A  its  value  found  from  (4), 

sin  a  =  tan  b  cot  B,  (7) 

and  similarly 

sin  b  =  tan  a  cot  A.  (8) 

From  8, 

tan  a  =  sin  6  tan  A 
6 


82  ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 

or 

sing  _  sin  h  sin  A 
cos  a  cos  A 

Substituting  in  this  equation  the  values  of  sin  a  and  cos  a 
found  from  5  and  1,  we  have 

sin  c  sin  A  cos  h  _  sin  h  sin  A 
cos  c  "      cos  A 

or 

cos  A  =  tan  h  cot  c,  (9) 

and  similarly 

cos  B  =  tan  a  cot  c.  (10) 

84.  Laws  of  Species  for  the  right  Spherical  Triangle.    We 

shall  prove  the  following  two  laws  of  species  for  the  right  spheri- 
cal triangle. 

1.  A  side  of  a  right  spherical  triangle  and  the  angle  opposite  it 
are  of  the  same  species. 

2.  If  the  hypotenuse  of  a  right  spherical  triangle  he  acute j  the 
two  other  sides  are  of  the  same  species,  and  if  the  hypotenuse  be 
obtuse,  the  two  other  sides  are  of  different  species. 

Proof  of  1. 

sin  b  =  tan  a  cot  A.  (8) 

Since  h  is  less  than  180°,  sin  b  is  positive.     Then  tan  a  and 

cot  A  are  of  the  same  sign,  and  hence  a  and  A  are  of  the  same 

species. 

Proof  of  2. 

cos  c  =  COS  a  cos  b.  (1) 

If  c  is  less  than  90°,  cos  c  is  positive.  Then  cos  a  and  cos  b 
are  of  the  same  sign,  and  hence  a  and  b  are  of  the  same  species. 
On  the  other  hand,  if  c  is  greater  than  90°,  cos  c  is  negative, 
then  cos  a  and  cos  b  are  of  opposite  signs,  and  hence  a  and  b 
are  of  different  species. 

85.  The  Solution  of  the  right  Spherical  Triangle.  When 
two  parts  of  a  right  spherical  triangle  are  given  the  remaining 
parts  may  be  computed  by  means  of  Napier's  rule  of  circular 
parts  and  the  laws  of  species.  The  method  in  detail  will  be 
illustrated  by  the  following  examples. 

When  a  relation  involves  a  trigonometric  function  of  an  obtuse 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.  83 

angle  we  replace  the  angle  by  its  supplement.  This  substitution 
in  any  one  of  the  formulae  (I)-(IO)  will  at  most  give  the  wrong 
sign  to  the  desired  function,  thus  leaving  it  uncertain  as  to 
whether  the  desired  angle  is  acute  or  obtuse.  The  laws  of 
species  are  intended  to  remove  this  ambiguity 
Ex.  1. 

a  =    47°  30'  40", 

c  =  120°  20'  30''. 

Applying  Napier's  rule  to  acA,  acB,  acb  and  solving  each  equa- 
tion for  the  unknown  part,  we  have 

.       sin  a  r>       X  J.  T      cos  c 

smA=  ~. — )      cos  ^  =  tan  a  cot  c,     cos  o  = 

sm  c  cos  a 

Since  c  is  obtuse  a  and  h  are  of  different  species.     Then  h  and 
B  are  obtuse,  and  A  is  acute. 

log  sin  a  =  9.86771  log  tan  a  =  0.03812  log  cos  c  =  9.70342 
log  sin  c  =  9.93603  log  cot  c  =  9.76740  log  cos  a  =  9.82959 
log  sin  A  =  9.93168  Tog  cos  5=  9.80552  log  cos  h  =  9.87383 
A  =  58°  41' 55"  5  =  180°  -  50°  16' 50"  6  =  180°  -  41°  35' 35" 
=  129°  43'  10"  =  138°  24'  25" 

Check:*  Napier's  rules  applied  to  A,  B,b  give 

cos  B  =  cos  h  sin  A. 

log  cos  h  =  9.87383 
log  sin  A  =  9.93168 
log  cos  B  =  9.80551 
Ex.  2.  a  =  103°  12',        A  =  97°  24'. 

Napier's  rules  give 

.    ,        ,  ,4         .    T)      cos  A        .  sin  a 

sm  0  =  tan  a  cot  A,     sm  B  = ,     sm  c  =  ~ — 7- 

cos  a  sm  A 

log  tan  a  =  0.62977  log  cos  A  =  9.10990    log  sin  a  =  9.98837 

log  cot  A  =  9.11353    log  cos  a  =  9.35860  log  sin  A  =  9.99637 

log  sin  b  =  9.74330  logTin  B  =  9.75130    log  sin  c  =  9.99200 

Since  a  is  obtuse  it  follows  that,  if  h  be  also  obtuse,  c  must 

*  To  test  the  accuracy  of  the  arithmetical  work  it  is  necessary  to  use  as 
a  check  some  formula  which  has  not  already  been  used.  In  any  right 
spherical  triangle  the  most  convenient  relation  for  this  purpose  is  that 
one  which  involves  the  three  required  parts. 


84 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY: 


be  acute,  and,  if  b  be  acute,  c  must  be  obtuse.  The  laws  of 
species  give  no  means  of  telling  whether  b  is  acute  or  obtuse. 
In  fact  there  are  two  triangles  with  the  given  parts  a  and  A, 

which  together  form  a  lune  of  angle 
A,  as  in  the  accompanying  figure. 
The  two  solutions  are  then 


(1) 


(2) 


b  =    33°  37'  25'', 
B  =    34°  20'    5", 
c  =  100°  58', 
6'  =  180°  -  b  = 


146°  22'  35", 


B'  =  180°  -  B  =  145°  39'  55", 


Fig.  57.  c'  =  180°  -  c  =    79°    2'. 

Check:  sin  6  =  sin  c  sin  B. 

log  sin  c  =  9.99200 

log  sin  B  =  9.75130 

log  sin  b  =  9.74330 
86.  Exercise  XV. 

Solve  the  following  spherical  triangles  in  which  C  =  90*^ 
1. 


3. 


a  = 

53°  48'  10", 

2. 

a  =  120°  40'  5", 

6  = 

73°  12'  20". 

c  =    78°  22'  25". 

a  = 

23°  59'  55", 

4. 

A  =    27°  54'  5", 

c  = 

133°  24'  50". 

b  =  100°  58'  10". 

b  = 

27°  50'  30", 

6. 

B  =    78°  53'  55", 

A  = 

114°  14'  25". 

b  =    40°  12'  10". 

A  = 

53°  54'  55", 

8. 

a  =  127°  54'  10", 

c  = 

127°  12'  10". 

A  =   105°  50'  25". 

A  = 

110°  13'  20", 

10. 

b  =    35°  55'  30", 

B  = 

73°  13'  35". 

c  =  143°  14'  10". 

a  = 

29°  13'  35",   ^ 

12. 

a  =  154°  42'  10", 

b  = 

54°  14'  25". 

c  =   148°  14'  20". 

7.  A  = 


11. 


87.  Quadrantal  Triangles.  A  spherical  triangle  is  said  to 
be  quadrantal  if  one  of  its  sides  is  equal  to  a  quadrant,  that  is, 
equal  to  90°. 

In  the  spherical  triangle  ABC  let  c  =  90°.  Then  in  the 
polar  triangle  A'B'C  we  have  C  =  90°.  When  any  two  parts 
of  the  quadrantal  triangle  in  addition  to  c  are  given,  this  triangle 
may  be  solved  by  the  methods  explained  in  the  foregoing 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  85 

articles;  we  may  find  the  unknown  parts  of  the  (right)  polar 
triangle,  and,  by  subtracting  these  from  180°,  find  the  required 
parts  of  the  quadrantal  triangle. 

Quadrantal  triangles  may  also  be  solved  directly  as  follows. 

Napier's  rule,  as  stated  for  the  right  triangle,  also  holds  for 
quadrantal  triangles  provided  we  take  as  the  circular  parts 
A,  B,  90°  -  a,  -  (90°  -  C),  90°  -  6. 

The  laws  of  species  for  quadrantal  triangles  are: 

1.  A  side  of  a  quadrantal  triangle  and  the  angle  opposite  it 
are  of  the  same  species. 

2.  If  the  angle  opposite  the  quadrant  he  acute,  the  two  other 
angles  are  of  different  species,  and  if  this  angle  he  ohtuse  the  two 
other  angles  are  of  the  same  species. 

The  proofs  of  these  laws  of  species  and  of  the  application  of 
Napier's  rule  to  quadrantal  triangles  are  left  as  exercises  to  the 
student. 
88.  Exercise  XVI. 

Solve  the  following  spherical  triangles  in  which  c  =  90°: 

1.    a  =  50°  49'  25",  2.  A  =  103°  25'  55", 

B  =  73°  12'  35".  C  =    79°  15'    5". 

3.    6  =  18°  28'  10",  4.    a  =  100°  13'  20", 

C  =  55°  58'  30"  A  =  123°  16'  45". 


CHAPTER  VIII. 
THE  SOLUTION  OF  OBLIQUE  TRIANGLES. 

89.  In  the  study  of  solid  geometry  we  have  learned  that 
two  triangles  are  equal  or  symmetrical  if 

1)  Three  sides  of  one  are  equal  respectively  to  three  sides  of 
the  other. 

2)  Three  angles  of  one  are  equal  respectively  to  three  angles 
of  the  other. 

3)  Two  sides  and  the  included  angle  of  one  are  equal  respec- 
tively to  two  sides  and  the  included  angle  of  the  other. 

4)  A  side  and  the  adjacent  angles  of  one  are  equal  respectively 
to  a  side  and  the  adjacent  angles  of  the  other. 

That  is,  if  the  three  parts  mentioned  in  any  one  of  these 
cases  be  given,  the  remaining  parts  are  fully  determined.  It 
is  then  natural  to  suppose  that,  given  such  a  set  of  three  parts, 
the  remaining  three  parts  may  be  computed.  In  addition  to 
these  four  cases  we  shall  find  that,  when  given  two  sides  and 
an  angle  opposite  one  of  them,  or  two  angles  and  a  side  opposite 
one  of  them,  there  will  be,  at  most,  two  distinct  triangles  having 
these  parts,  and  that  the  remaining  parts  of  each  triangle  may 
be  computed. 

90.  We  shall  now  show  how  to  find  the  unknown  parts  of 
a  spherical  triangle  of  which  we  are  given 

1)  The  three  sides,  a,  h,  c. 

2)  The  three  angles,  A,  B,  C. 

3)  Two  sides  and  the  included  angle,  e.  g.,  a,  b,  C. 

4)  A  side  and  the  two  adjacent  angles,  e.  g.,  c,  A,  B. 

5)  Two  sides  and  the  angle  opposite  one  of  them,  e.  g.,  a,  &,  A. 

6)  Two  angles  and  the  side  opposite  one  of  them,  e.  g.,  A,  5,  a. 

91.  Case  (1).     Given  the  Three  Sides,  a,  h,  c. 

The  half  angles  may  be  determined  by  the  formulae  (IV), 
page  77.  It  is  important  to  observe  that,  since  the  angles  are 
less  than  180°,  each  of  the  half  angles  is  less  than  90°.  It 
therefore  is  not  necessary  to  use  the  laws  of  species.     After 

86 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.  87 

finding  the  three  angles  the  accuracy  of  the  results  should  be 
tested  by  substituting  the  angles  in  some  relation  that  has  not 
already  been  used.  For  this  the  sine  proportion  (III),  page  75, 
is  the  most  convenient. 

Ex.  1.  a  =  58°,     h  =  80°,     c  =  96°. 

Formulae : 

^     A  K  ^     B  K  ^     C  K 

tan  —  =  -. — ; r,     tan  -7^  =  - — 7 — r^ ,      tan  7^  = 


sin  (s— a)'  2       sin  (s  — 6)'  2       sin  (s—cy 

K  =    fsin  (g  —  o)  sin  (s  —  h)  sin  {s  —  c) 
\  sin  s 

a  =    58° 
b  =    80°        log  sin  (s  -  a)  =  9.93307  A 

96°        log  sin  {s-h)=  9.77946     ^^g  tan  ~  =  9.72542 


c  = 


2s  =  234°        log  sin  {s  -  c)   =  9.55433 


B 


s  =  117°  9.26686     log  ism—  =  9.87903 

s  -  a  =    59°  log  sin  g     =  9.94988  ^ 

s  -h  =    37°  log  X2        =  9.31698     log  tan -^  =  0.10416 

s  -  c  =    21°  log  K         =  9.65849 
s  =  117° 

~  =  27°  59'  10",        A  =  55°  58'  20'', 
^  =  37°  7'  20",  5  =  74°  14'  40", 

~  =  51°  48'  20",         C  =  103°  36'  40". 

^,     ,  ,      sin  A       sin  5      sin  C 

Check :  a  =  — =  -; — r  =  -. . 

sm  a       sm  0       sm  c 

log  sin  A  =  9.91843    log  sin  B  =  9.98337    log  sin  C  =  9.98763 

log  sin  a  =  9.92842     log  sin  h  =  9.99335     log  sin  c  =  9.99761 

log  d  =  9.99001    ■       log  d  =  9.99001  log  d  =  9.99002 

Ex.  2.     a  =  53'  12'  35",    h  =  75°  14'  25",    c  =  69°  27'  20". 

92.  Case  (2).     Given  the  Three  Angles,  A,  B,  C. 

Since  the  angles  of  a  spherical  triangle  are  the  supplements 
of  the  corresponding  sides  of  the  polar  triangle,  we  may  find 
the  sides  of  the  polar  triangle.  Then,  by  the  method  of  case 
(1),  we  may  compute  the  angles  of  the  polar  triangle.  The 
supplements  of  these  angles  will  then  be  the  required  sides  of 
the  original  triangle. 


88  ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 

Ex.  1.  A  =  78°  40'  log  sin  (s  -  a)'  =  9.98338 

B  =  63°  50'  log  sin  (s  -  6/  =  9.93495 

C  =  46°  20'  log  sin  (s  -  c)'  =  9.82481 

.       ,  9.74314 

In  the  polar  triangle  .      ,      o  oo«k.i 

//  =  101°  20'  log  sm  s'  =  8.88654 

?,  _l1.o-,n'  log  K'^  =  0.85660 

,.  =  133040^  log  Z'=  0.42830 

2s-^^350^'  logtan4-'  =  0.44492, 

g/  _  X75°  35'  -^ 

^  =  70°  15'  10"-,  A'  =  140°  30'  20"-, 

~  =  72°  11'  50"+,  B'  =  144°  23'  40"+, 

£1 

~  =  76°  0'  30"-,  C  =  152°  1'-. 

Hence,  in  the  original  triangle, 

a  =  39°  29'  40",     h  =  35°  36'  20",     c  =  27°  59'. 

^i_    ,  sin  A      sin  B      sin  C 

Check: =  ~. — r  =  -^ . 

sm  a       sm  0       sm  c 

log  sin  A  =  9.99145    log  sin  B  =  9.95304    log  sin  C  =  9.85936 

log  sin  a   =  9.80346    log  sin  b  =  9.76507     log  sin  c  =  0.67137 

0.18799  0.18797  0.18798 

Ex.  2.    A  =  105°  14'  20",  B  =  80°  0'  10",  C  =  68°  23'  35". 

93.  Case  (3) .  Given  Two  Sides  and  the  Included  Angle,  e.  g., 
a,  h,  C, 

Solving  the  relations  (c),  (d),  and  (6),  Art.  79,  for 
tan  i(A  -\-B),  tan  i{A  —  B),  and  tan  c/2  respectively,  we  have 

(J 

cos  i{a  —  h)  cot^ 

tan  |(A  +  B)  = tt — r-^^r , 

^  cos  -JCa  +  h)      ' 

C 

sin  ^(a  —  6)  cot  ^ 

tan  UA  —  B)  = r—^. — ^-r^ — » 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.  89 

,  .       c       sin  i(A  +  B)  tan  -^(a  —  h) 

and  tan  ^  = -. — ^Ta d\ • 

2  sin^(A  —  B) 

By  the  first  two  of  these  relations  we  may  compute  half  the 
sum,  and  half  the  difference  of  the  unknown  angles,  and  so  can 
find  these  angles.  This  being  done,  the  third  side,  c,  may  be 
found  by  means  of  the  last  relation  above.  The  accuracy  of 
the  results  may  then  be  tested  by  the  sine  proportion. 

Observe  that  ^{A  —  B)  and  c/2  are  always  acute,  and  that 
therefore  the  law  of  species  need  not  be  used  in  finding  the 
values  of  these  angles.  On  the  other  hand,  \(A  -\-  B)  may  be 
obtuse.  In  fact,  this  angle  is  acute  or  obtuse  according  as 
^{a  +  h)  is  acute  or  obtuse,  page  79. 

Ex.  1: 


c  =    40°  20' 

|(a  +  c)  =  70°  25' 

a  =  100°  30' 

i(a  -  c)  =  30°    5' 

B  =    46°  40' 

B/2  =  23°  20' 

Formulae: 

cos  ^{a  —  c)  cot  -X 

tan  i(A  +  C)  = ,  ,    ,     ,     , 

^^  ^  cos  I  (a+  c)      ' 

sin  I  (a  —  c)  cot  — 
taniU-C)=      ,in^(^  +  ,)      , 

h  _  sin  |(A  +  C)  tanj-  (a  —  c) 
^^2  ~  sin  i{A  -  C) 

log  cos  i(a  -  c)  =  9.93717         log  sin  i{a  -  c)  =  9.70006 
log  cot  J5/2  =  0.36516  0.36516 

0.30233  0.06522 

log  cos  iia  +  c)  =  9.52527  log  sin  i{a  +  c)  =  9.97412 
log  tan  i(A  +  C)  =  0.77706  log  tan  i{A  -  C)  =  0.09110 
i(A  +  C)  =  80°  30'  50"+.  i(A  -  C)  =  50°  58'-. 

Hence,  A  =  131°  28'  50", 

C  =    29°  32'  50". 
log  sin  i(A  +  C)  =  9.99402 
log  tan  |(a  -  c)  =  9.76290 
9.75692 
log  sin  i(A  -  C)  =  9.89030 
log  tan  6/2  =  9.86662 


90  ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 

6/2  =  36°  20'  15''- 
and  hence,  b  =  72°  40'  30"-. 

^,     ,  sin  A      sin  B      sin  C 

Check:  - —  =  ^—7  =  -. . 

sm  a      sm  0       sin  c 

log  sin  A  =  9.87459    log  sin  B  =  9.86176    log  sin  C  =  9.69297 

log  sin  a  =  9.99267     log  sip  6  =  9^984     log  sin  c  =  9.81106 

9.88192  9.88192  9.88191 

Ex.  2.    h  =  109°  12'  10",  c  =  131°  18'  25",  A  =   46°  14'  55". 

94.  Case  (4).  Given  a  Side  and  the  Two  Adjacent  Angles, 
e.  g.,  c,  A,  B. 

The  solution  in  this  case  is  exactly  similar  to  that  of  III. 

Ex.  1:  Formulse: 

C=110°40'  j,^      p.,      a 

B  =  100°36'         ,      ,,   ^,,      cosKC-B)tan- 

a   =    76°  38'         ^^^  *^^  +  ^)  =  "cosKC  +  5)  " ' 
|((7  +  5)  =  105°  38'  sin  \{C  -  B)  tan  ^ 

i(C  -  B)  =     5°   2'     tan  4(0  -  6)  =  -,i,x(c  +  ^)      > 

1  =  38°  19'  cot  4  =  BJrLKc+6)tani(C-B)^ 

2  2  sin  \{c  —  0) 

log  cos  i(C  -  B)  =  9.99832  log  sin  \(C-  B)  =  8.94317 

log  tan  a/2  =  9.89775  9.89775 

9.89607  8.84092 

log  cos  4(C  +  5)  =  9.43053  log  sin  i(C  +  5)  =  9.98363 

log  tan  -i(c  +  &)  =  0.46554  log  tan  Kc  -  &)  =  8.85729 

i(c  +  6)  =  180°  -  71°  6'  5"+        \{c  -  h)  =4°7'5"- 

=  108°  53'  55"- 

c  =  113°  1'-,     6  =  104°  46'  50". 

log  sin  |(c  +  6)  =  9.97593 

log  tan  \{C  -  B)  =  8.94485 

8.92078 

log  sin  ijc  -h)  =  8.85617 

log  cot  A/2  =  0.06461 

4  =  40°  45'  15"  - ,    A  =  81°  30'  30"  - . 

^,     ,  sin  A      sin  B      sin  C 

Check:  —. =  — — 7  =  ~. — • 

sin  a       sm  o       sm  c 


ELEMENTS   OF   SPHERICAL  TRIGONOMETRY.  91 

log  sin  A  =  9.99521    log  sin  B  =  9.99252    log  sin  C  =  9.97111 

log  sin  a  =  9.98807     log  sin  b  =  9.98539     log  sin  c  =  9.96397 

0.00714  0.00713  0.00714 

Ex.  2.  A  =  59°  19'  15'',  B  =  76°  14'  15",  c  =  130°  14'  50". 
95.  Case  (5).     Given  two  Sides  and  the  Angle  Opposite  One 
of  Them,  e.  g.,  a,  6,  A. 
From  III,  page  75, 

.    ^      sin  &  sin  A 

smij  = -. . 

sm  a 

We  may  then  compute  the  value  of  log  sin  B.  But,  since 
sin  B  =  sin  (180°  —  5),  we  must  apply  the  law  of  species  to  tell 
whether  B  is  acute  or  obtuse.  Denote  the  acute  angle,  found 
in  the  tables  from  the  value  of  log  sin  B,  by  B,  and  the  corre- 
sponding obtuse  angle  ( =  180°  —  B)  by  5'.  If  B  is  to  be  a 
possible  solution,  the  law  of  species,  page  79,  states  that  i(a  +  6) 
and  ^{A  +  B)  must  be  of  the  same  species.  Similarly,  if  B' 
is  a  possible  solution,  |(a  +  6)  and  -1{A  +  5')  must  be  of  the 
same  species.  Hence,  this  law  will  show  whether  there  be  one 
solution,  two  solutions,  or  no  solution. 

Having  found  B,  we  may  compute  the  remaining  parts,  c  and 
C,  by  Napier's  analogies  (6)  and  {d),  page  78.  We  may  then 
check  by  the  sine  proportion. 

Ex.    1.    a  =  30°  20',        h  =  46°  30',       A  =  36°  40'. 

sin  h  sin  A 


Formulae:  sin  B  = 


sm  a 


C  ^  sin  ^{h  +  a)  tan  jjB  -  A) 
^^^2  smiib-a) 

c  _  sin  i{B  -\-  A)  tan  |(b  —  a) 
^^2  "  sin  i{B  -  A)  • 

log  sin  b  =  9.86056 

log  sin  A  =  9.77609 

9.63665 

log  sin  a  =  9.70332 

log  sin  B  =  9.93333 

B  =  59°  3'  30",  B'  =  180°  -  59°  3'  30" 

=  120°  56'  30". 


92 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 


i(6  +  a)  =  38°  25'  i(5  -  A)  =  11°  11'  45'' 

i(6  -a)  =    8°    5',  i(B'  +  A)  =  78°  48'  15" 

i{B  +  A)=  47°  51'45",        i(5'  -  A)  =  42°    8'  15" 
There  are  two  solutions  since  both  |(5  +  A)  and  |(B'  +  A) 
are  of  the  same  species  as  |-(6  +  a). 

First  Solution.  Second  Solution, 

log  sin  i{h  +  a)  =  9.79335  9.79335 

log  tan  i{B  -  A)  =  9.29651  9.95653 


9.08986 

log  sin  i{h  -  a)  =  9.14803 

log  cot  C/2  =  9.94183 

^  =  48°  49' 

C  =  97°  39' 
log  sin  i(B  +  A)  =  9.87013 
log  tan  4(6  -  a)  =  9.15236 


30" 


14°  2'  35"- 


9.02249 
9.28817 
log  tan  c/2  =  9.73432 


log  sin  i(B  -  A) 


~  =  28°  28'  30' 


9.74988 

9A4803 

0.60185 
C 
2 

C'=28°5'  10"- 

9.99166 

9.15236 

9.14402 

9.82666 

9.31736 

^  =  11°  43'  55"  - 
c'  =  23°  27' 50"  -. 


c  =  56°  57' 
Hence  the  solutions  are 

(1)  B  =  59°    3'  30"  (2)  B'  =  120°  56'  30" 

C  =  97°  39'  C  =    28°    5'  10"  - 

c  =  56°  57'  +  c'  =    23°  27'  50"  - 

p,     ,  ^  sing  _  sinjC  _  sin  C 

sin  b      sin  c      sin  c'  ' 

log  sin  B  =  9.93333   log  sin  C  =  9.99612   log  sin  C  =  9.67284 

log  sin  b  =  9.86056   log  sin  c  =  9.92335   log  sin  c'  =  9.60007 

0.07277  0.07277  0.07277 

Ex.  2.    6  =  32°  18'  10",  c  =  50°  14'  15",   C  ==  48°  12'  10". 

96.  Case  (6).  Given  Two  Angles  and  the  Side  Opposite 
One  of  Them,  e.  g.,  A,  B,  a.  The  solution  of  this  case  is 
exactly  similar  to  that  of  Case  V. 

*  It  is  not  necessary  to  use  the  ratio  sin  A  Ism  a  since  this  was  used  in 
the  solution. 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 


93 


Ex.  1. 

B  =  69°,          C  =  132°,          b  =  65°. 

Formulae: 

sin  C  sin  h 

sm  c  = : — 5 — , 

sm  B 

A      sin  i{c-\-h)  ism  UC  -  B) 

^^^  2  ~               sin  Kc  -  ?>) 

a       sin  -KC'  +  5)  tan  -i(c  -  6) 

^^2"              sin  UC-B) 

log  sin  C  =  9.87107 

log  sin  h  =  9.95728 

9.82835 

log  sin  B  =  9.97015 

T'V.i-i'r* 

log  sin  c  =  9.85820 

inen, 

c  = 

=  46°  10'  25''            c'  =  180°  -  46°  10'  25" 

=  133°  49'  35" 

-i(c'  +  h)  =  99°  24'  50"  - 
4(c'  -h)  =  34°  24'  50"  - 


and 

^(C  +  B)  =  100°  30' 
i(C  -  B)  =  31°  30' 
i(c  +  6)  =    55°  35'  15"  - 

Of  the  two  values,  c  and  c',  the  latter  alone  is  a  solution  since 
it  only  obeys  the  law  of  species.  Then,  as  there  can  be  but 
one  solution,  we  may  omit  accents  and  take  c  =  133°  49'  35". 
Then, 

i(c  +  6)  =  99°  24'  50"  - 
i(c  -h)  =  34°  24'  50"  - 

log  sin  i(C  +  B)  =  9.99267 


|(C  +  B)  =  100°  30' 

i(C  -  B)  =    31°  30' 

log  sin  i(c  +  6)  =  9.99411 

log  tan  i(C  -  B)  =  9.78732 

9.78143 

log  sin  -i(c  -  6)  =  9.75218 

log  cot  A/2  =  0.02925 

I  =  43°  4'  20", 

A  =  86°  8'  40", 


log  tan  |(c  -  6)  =  9.83573 

9.82840 

log  sin  i(C  -  B)  =  9.71809 

log  tan  a/2  =  0.11031 

=     52°  12'  -, 


a 


Check: 


a  =  104°  24'- 


sin  A      sin  C 


sm  a 
log  sin  A  =  9.99902 
log  sin  a  =  9.98614 
0.01288 


sm  c 
log  sin  C 
log  sin  c 


9.87107 
9.85820 
0.01287 


94  ELEMENTS  OF   SPHERICAL  TRIGONOMETRY.       ' 

Ex.  2.     A  =  70°  14'  15'^  B  =  59°  12'  25'',  b  =  51°  18'  35' 
97.  Exercise  XVI.     Solve  the  following  spherical  triangles. 


3.  A  = 


5.    a  = 


7.    a  = 


9.  A  = 


11.    c  = 


13.  A  = 


15.  A  = 


17. 


a  = 

48°  35' 

2. 

A  = 

105°  24'  10" 

b  = 

77°  23' 

b  = 

110°  5'  15" 

c  = 

80°  56' 

c  = 

80°  37'  35" 

A  = 

120°  56'  25" 

4. 

a  = 

29°  15'  25" 

B  = 

80°  32'  10" 

c  = 

50°  29'  55" 

C  = 

100°  27'  50" 

B  = 

47°  48'  50" 

a  = 

103°  22'  50" 

6. 

a  = 

58°  32'  50" 

B  = 

76°  13'  25" 

b  = 

42°  23'  10" 

C  = 

37°  58'  55" 

A  = 

60°  36'  35" 

a  = 

102°  14'  10" 

8. 

A  = 

70°  25'  10" 

b  = 

74°  14'  35" 

B  = 

50°  46'  25" 

c  = 

118°  29'  25" 

C  = 

80°  39'  55" 

A  = 

105°  35'  35" 

10. 

A  = 

123°  34'  50" 

B  = 

120°  23'  10" 

B  = 

78°  22'  35" 

c  = 

74°  24'  25" 

b  = 

98°  29'  25" 

c  = 

73°  14'  10" 

12. 

A  = 

58°  58'  55" 

b  = 

60°  12'  10" 

B  = 

36°  49'  45" 

B  = 

40°  28'  35" 

C  = 

96°  37'  35" 

A  = 

167°  43'  35" 

14. 

a  = 

76°  56'  55" 

B  = 

103°  25'  25" 

c  = 

48°  48'  50" 

c  = 

70°  45'  50" 

B  = 

39°  32' 35" 

A  = 

80°  54'  55" 

16. 

a  = 

47°  54'  55" 

B  = 

90° 

b  = 

36°  25'  50" 

c  = 

47°  16'  10" 

c  = 

80°  19'  10" 

b  = 

54°  51'  35" 

18. 

B  = 

54°  51'  25" 

a  = 

79°  22'  25" 

A  = 

79°  22'  25" 

C  = 

28°  29'  55" 

c  = 

28°  29'  55" 

98.  Geographical  Problems. 

The  position  of  a  point  on  the  earth's  surface  is  determined 
if  the  latitude  and  longitude  of  the  point  be  given.  By  the 
aid  of  spherical  trigonometry  we  may  find  the  distance  between 
two  points  whose  latitudes  and  longitudes  are  known,  and  also 
the  bearing  of  either  point  from  the  other.  Since  the  shortest 
distance  between  two  points  is  along  the  arc  of  a  great  circle, 
a  navigator  sails  as  nearly  as  possible  upon  the  great  circle  arc 
from  the  point  of  departure  to  the  desired  destination.     The 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 


95 


course,  when  sailing  upon  a  great  circle  arc,  is  continually 
changing  except  in  the  special  cases  of  sailing  along  the  equator 
or  a  meridian.  If,  on  the  other  hand,  a  ship  kept  upon  the 
same  course,  for  example  N.  25°  E.,  it  would  sail  in  a  spiral 
slowly  approaching  one  of  the  poles.  Such  a  path  upon  the 
earth's  surface  is  known  as  a  loxodrome  or  rhumb  line. 

99.  In  the  following  table  the  latitudes  and  longitudes  of 
several  cities  are  given: 

Baltimore  39°  17'  N.,  76°  37'  W. 
Boston  42°2rN.,    71°    4' W. 

Cape  Town  33°  56'  S.,  18°  26'  E. 
Halifax  44°  40' N.,    63°  35' W. 

Honolulu  21°  18'  N.,  157°  55'  W. 

Liverpool  53°  24' N.,  3°  4' W. 
New  York  40°  43' N.,  74°  W. 
San  Francisco  37°  48'  N.,  122°  24'  W. 

100.  Exercise  XVII.  In  the 
following  problems  the  earth 
is  assumed  to  be  spherical,  and 
the  radius  is  taken  as  3960 
miles. 

1.  Find  the  distance  from 
Halifax  to  Liverpool,  and  the 
bearing  of  each  place  from  the 
other.  Also  find  the  course  of 
a  ship  sailing  from  Halifax  to 
Liverpool  when  it  crosses  the 
meridian  55°  W.,  the  latitude 
of  this  point,  and  the  distance 
the  ship  has  sailed. 

Let  G,  L  and  H  denote  respectively  Greenwich,  Liverpool, 
and  Halifax.     Then 

GNL  =    3°    4',     LN  =  90°  -  53°  24'  =  36°  36', 
GNH  =  63°  35',     HN  =  90°  -  44°  40'  =  45°  20'. 
.-.  LNH  =  60°  31'. 

Then  we  are  given  two  sides  and  the  included  angle  of  the 
triangle  HNL.     We  can  then  solve  for  the  angles  NHL  and 


96  ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 

HLN  and  for  the  side  HL  by  the  method  of  Art.  93.     This  gives 

NHL  =  54°  54'  20'', 
HLN  =  77°  25'  25", 
HL  =  39°  22'    5"  =  39.368°. 

Then  the  distance  from  Halifax  to  Liverpool  is 

,      39.368  X  TT  X  3960       __,      ..  ,.    .   _, 

d  = TgQ =  2721  miles.  [Art.  66] 

The  bearing  of  Halifax  from  Liverpool  is  N.  77°  25'  25",  W., 
and  that  of  Liverpool  from  Halifax  is  N.  54°  54'  20"  E. 

To  solve  the  last  part  of  the  problem  consider  the  triangle 
HNA,  where  AN  is  the  meridian  of  55°  W. 

In  this  triangle 

HN  =  45°  20', 
NHA  =  54°  54'  20", 
HNA  =  63°  35'  -  55°  =  8°  35'. 

We  have  then  a  side  and  the  two  adjacent  angles,  and  can 
therefore  solve  for  the  remaining  parts  by  the  method  of  Art. 
94.     Hence. 

AN  =  41°  38'  20", 
HAN  =  118°  51'  25", 

HA  =  6°  57'  40"  =  6.961°. 

Therefore  the  latitude  of  the  point  is  48°  21'  40",  the  course  is 
N.  180°  -  118°  51'  25"  E.,  or  N.  61°  8'  35"  E.,  and  the  distance 
sailed  is  given  by 

,      6.961  X  TT  X  3960        .^,    .      ., 
a  = :r^ =  481  +  miles. 

2.  A  ship  sails  from  Halifax  to  Liverpool  along  the  arc  of  a 
great  circle.  Find  her  latitude  and  longitude  after  she  has 
sailed  one  thousand  miles.     Find  also  her  course  at  this  point. 

3.  Find  the  distance  from  New  York  to  San  Francisco. 

4.  A  ship  sails  from  San  Francisco  to  Honolulu.  Find  the 
course  she  is  steering  when  she  has  covered  half  the  distance. 

5.  A  ship  sails  from  Boston  to  Cape  Town.  Find  the  total 
distance  sailed,  and  her  course  when  1500  miles  from  Cape 
Town. 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY.  97 

6.  After  sailing  1000  miles  from  Halifax  a  ship  crosses  the 
parallel  of  latitude  of  40°.  What  will  be  the  position  of  the 
ship  (that  is,  the  latitude  and  longitude)  after  she  has  sailed 
800  miles  further  on  the  same  great  circle. 

7.  Find  the  area  in  square  miles  of  the  triangle  whose  vertices 
are  Baltimore,  Cape  Town,  and  Liverpool. 


CHAPTER  IX. 
OTHER   FORMULiE   RELATING   TO   SPHERICAL   TRIANGLES. 
101.  From  IV,  page  77,  we  have 

tan  7:  =  -r-7 ^r.  (1) 

2       sin(s  —  a) 

Since  this  relation  is  true  for  any  spherical  triangle  it  is  true 
for  the  polar  triangle  of  a  given  triangle.  We  may  then  replace 
A'  by  180°  -  a,  a'  by  180°  -  A,  etc. 

Let2S  =  A-\-  B  +  C,  so  that  s'  =  1(540°  -  [A -\- B  +  C]) 
=  270°  -  S. 
Then  j^{-  A  -\-  B  +  C)  =  S  -  A,  etc. 

Hence 

s'  -  a'  =  90°  -  {S  -  A), 

Similarly  s'  -  h'  =  90°  -  {S  -  B), 

and  s'  -  c'  =  90°  -  {S  -  C). 

Therefore 


^,  ^      Isin  (s'  —  a')  sin  (s'  —  bQ  sin  js'  —  c') 
\  sin  s' 

/cos  (S  -  A)  cos  (>S  -  g)  cos  {S  -  C) 

^i  -  coss 

Now  let 


,  ,  —  cos  S 


=4 


(cos  {S  -  A)  cos  {S  -  B)  cos  {S  -  C) 
so  that 

«■ = I- 

The  equation  (1)  then  becomes 

a  1 

cot  -  = 


2      /b  cos  (^  -  A) 

98 


ELEMENTS  OF  SPHERICAL  TRIGONOMETRY. 


99 


or 


Similarly 


tan^  =  k  cos  {S  —  A). 
tan  ^  =  A;  cos  (*S  —  5), 


VI 


and 


tan  »  =  A;  cos  {S 


C). 


By  means  of  these  equations  a  triangle  in  which  the  three 
angles  are  given  may  be  solved  directly. 

102.  Radius  of  the  Circumscribed  Circle.  Let  Ai,  Bi,  and  Ci 
be  the  middle  points  of  the  sides  of  the  spherical  triangle  ABC. 
At  Ai  draw  the  arc  of  a  great  circle  perpendicular  to  the  side  BC, 
and  similarly  draw  arcs  through  Bi  and  Ci  perpendicular  to  the 
corresponding  sides  of  ABC. 
These  three  arcs  will  meet  in 
a  point  0.  [This  may  be 
proved  in  a  manner  entirely 
analogous  to  that  employed 
for  the  corresponding  theo- 
rem in  plane  geometry.] 
Draw  the  great  circle  arcs 
0 A,  0 B  and  OC.  The  right 
triangles  OAiB  and  OAiC 
are  symmetrical  since  BAi  = 
AiC  and  OAi  is  common  to 
the  two  triangles.  Hence 
the  angles  AiBO  and  AiCO 
are  equal.     Similarly  BiAO  ■■ 

Now 
2S  =  A  +  B  +  C, 

=  {OABi  +  OACi)  +  (OBAi  +  OBCi)  +  {OCAi  +  OCB,), 

=  (OABi  +  OCBO  +  (OACi  +  OBCi)  +  {OBAi  +  OCAi), 

=  20ABi  +  20ACi  +  20BAi, 

=  2A+  20BAi. 
Therefore  OBAi  =  S  -  A. 

Then  in  the  right  triangle  OBAi  we  have  OBAi  =  S  —  A, 
BAi  =  a/2  and  OB  =  r,  where  r  is  the  radius  of  the  circum- 
scribed circle.     Applying  Napier's  rule  to  this  triangle,  we  have 


Fig.  59. 


BiCO,SindCiAO=CiBO. 


100 


ELEMENTS  OF   SPHERICAL  TRIGONOMETRY. 


or, 


cos  OBAi  =  cot  OB  tan  AiB, 
cos  {S  —  A)  =  cot  r  tan  -. 


Substituting  for  tan  a/2  its  value  from  VI,  Art.  101,  we  have 

cos  {S  —  A)  =  cot  r-k  cos  {S  —  A). 
Hence  it  follows  that 


tanr 


=  fc  =  ^ 


—  cos/S 


cos  {S  -  A)  cos  {S  -  B)  cos  {S  -  C) 


VII 


By  this  formula  we  may  find  the  radius  r,  that  is  the  polar 
distance,  of  the  small  circle  circumscribed  to  the  spherical 
triangle. 

103.  Radius  of  the  In- 
scribed Circle.  Draw  arcs 
of  great  circles  bisecting  the 
angles  of  the  spherical  tri- 
angle. These  arcs  meet  in 
a  point  0.  Through  0  draw 
the  great  circle  arc  OAi  per- 
pendicular to  the  side  BC, 
and  similarly  OBi  and  OCi 
perpendicular  to  CA  and  AB 
respectively.  The  right  tri- 
angles OACi  and  OABi  are 
equal  since  the  angles  OACi 
and  OABx  are  equal  and 
the  hypotenuse  OA  is  common.  Then  ABi  =  ACi.  Similarly 
BCi  =  BAi  and  CAi  =  CBi. 
Now, 

2s  =  a  +  5  +  c, 

=  {BA,  +  A^C)  4-  (CBi  +  B,A)  +  {ACi  +  C^B), 
=  {BA,  +  BCi)  +  {A^C  +  CB,)  +  {B,A  +  C,A), 
=  2BAi  +  2AiC  +  2BiA, 
=  2a  +  2BiA. 
Then  5iA  =  s  -  a. 

We  then  have  in  the  right  triangle  OBiA,  BiA  =  s  —  a,  OABi 
=  A/2  and  05i  =  R,  where  72  is  the  radius  of  the  inscribed 


ELEMENTS  OF   SPHERICAL   TRIGONOMETRY.         101 

circle.     Applying  Napier's  rule  to  this  triangle  we  have 

A 

sin  {8  —  a)  =  tan  R  cot—, 

or 

A 

tan  R  =  sin  (s  —  a)  tan  — ,  (1) 


=  sin  (s  —  a)  - 


2' 
K 


sin  (s  —  a)' 
Therefore, 


tan  72  =  Z  =  ^Mn(s-a)sin(s-b)sin(8-c)^    yjjl 
\  sin  s 

104.  Escribed  Circles.  Let  us  produce  the  sides  AB  and 
AC  of  the  spherical  triangle  ABC  to  meet  again  in  A\,  The 
circle  inscribed  in  the  triangle  AiBC  is  said  to  be  an  escribed 
circle  of  ABC.  Denote  the  radius  of  this  circle  by  R^.  Ob- 
viously there  are  two  other  escribed  circles  of  ABC.  The  radii 
of  these  circles  will  be  denoted  by  Rj^  and  R^. 

Applying  equation  (1)  of  the  last  article  to  the  triangle 
AiBC  J  we  have 

tan  72,  =  tan ^  sin  [i(a  +  180°  -b  +  180°  -  c)  -  a], 


2 
K 


sin  (s  —  a) 
K  sin  s 


sin  [180°  -  \{a  +  h-\-  c)], 


Hence, 


sin  (s  —  a)' 

K  sin  s 


tan  R^  = 


sin  (s  —  ay 


,       „           K  sin  s  -.„ 

tan  Rj^  =  - — -. TT,  IX. 

sm  (s  —  h) 

,       ^  i^  sin  s 

tan  R„  = 


sin  (s  —  c) 


TABLE  I, 


LOGARITHMS 

OF 

NUMBERS  FROM 
1  TO   10,000. 


103 


104 


N.     L.      0 


100 


130 

131 
132 
133 
134 
135 
136 
137 
138 
139 
140 
141 
142 
143 
144 
145 
146 
147 
148 
149 
150 


00  000 


.  432 

^860 

J  284 

)703 

J  119 

531 

1 938 

<342 

1 743 

139 

532 

922 

308 

690 

070 

446 

819 

188 

555 

918 

279 

636 

991 

342 

691 

037 

12  1^27 
1^  ^  057 

12  1^^^ 

^^  ^  033 
354 

13  1^^2 
U  [988 
^*  ^  301 


00 
01 
01 
02 

02 
03 
03 
04 

04 
05 
05 
06 

06 
07 

07 
08 

08 
09 
09 
10 


17 


613 
922 
229 
534 
836 


1 

-^137 
435 
732 
026 
319 
609 


N.      L.     0 


043 


475 
903 
326 
745 
160 
572 
979 
383 
782 


179 


571 
961 
346 
729 
108 
483 
856 
225 
591 


954 


314 
672 
'026 
377 
726 
072 
415 
755 
093 


428 


760 
090 
418 
743 
066 
386 
704 
^019 
333 


644 


953 
259 

564 
866 
167 
465 
761 
056 
348 
638 


218 


610 
999 

385 
767 
145 
521 
893 
262 
628 


990 


350 
707 
*061 
412 
760 
106 
449 
789 
126 


461 


793 
123 
450 
775 
098 
418 
735 
*051 
364 


675 


983 
290 
594 
897 
197 
495 
791 
085 
377 
667 


130 


561 
988 
410 
828 
243 
653 
*060 
463 
862 


258 


650 
^038 
423 
805 
183 
558 
930 
298 
664 


=027 


386 
743 

*096 
447 
795 
140 
483 
823 
160 
494 
826 
156 
483 
808 
130 
450 
767 

*082 
395 


706 


'014 
320 
625 
927 
227 
524 
820 
114 
406 
696 


173 


604 
*030 
452 
870 
284 
694 
*100 
503 
902 
297 


689 
W7 
461 
843 
221 
595 
967 
335 
700 


*063 
422 
778 

*132 
482 
830 
175 
517 
857 
193 


528 


860 
189 
516 
840 
162 
481 
799 
*114 
426 


737 


^045 
351 
655 
957 
256 
554 
850 
143 
435 
725 


217 


647 
^072 
494 
912 
325 
735 
141 
543 
941 


336 


727 
=115 
500 
881 
258 
633 
=004 
372 
737 


099 


458 
814 
167 
517 
864 
209 
551 
890 
227 


561 


893 
222 
548 
872 
194 
513 
830 
=145 
457 
768 


=076 
381 
685 
987 
286 
584 
879 
173 
464 
754 


260 


689 

ni5 

536 
953 
366 
776 
181 
583 
981 


376 


766 
154 
538 
918 
296 
670 
*041 
408 
773 


135 


493 
849 
*202 
552 
899 
243 
585 
924 
261 


594 


926 
254 
581 
905 
226 
545 
862 
176 
489 
799 


=106 
412 
715 
=017 
316 
613 
909 
202 
493 
782 


303 


732 

*157 

578 

995 

407 

816 

*222 

623 

*021 


415 


805 
192 
576 
956 
333 
707 
=078 
445 
809 


171 


628 


959 

287 
613 
937 

258 
577 
893 
=208 
520 


829 


137 
442 
746 
=047 
346 
643 
938 
231 
522 
811 


346 


775 
*199 

620 
*036 

449 

857 
*262 

663 
*060 

454 


844 
^231 
614 
994 
371 
744 
115 
482 
846 


^207 


565 
920 
*272 
621 
968 
312 
653 
992 
327 


661 


992 
320 
646 
969 
290 
609 
925 
=239 
551 


860 


168 
473 
776 
*077 
376 
673 
967 
260 
551 
840 


389 


817 
*242 

662 
*078 

490 

898 
*302 

703 
*100 


493 


S83 
^269 
652 
^032 
408 
781 
151 
518 
882 


243 


600 
955 

*307 
656 

*003 
346 
687 

*025 
361 


694 


'024 
352 
678 
=001 
322 
640 
956 
=270 
582 


891 


198 
503 
806 
*107 
406 
702 
997 
289 
580 
869 


P.P. 


44 

4-4 
8.8 
13.2 
17.6 
22.0 
26.4 
30.8 
35-2 
39-6 


43 


42 


17.2  16.8 
21.5  2r.o 
25.8  25  2 
30.1  29.4 
34-4133-6 
38-737-S 


12.3 
16.4 
20.5 
24.6 
28.7 
32.8 
9 '36.9 


40 

4.0 
8.0 
12.0 
16.0 
20.0 
24.0 
28.0 
32.0 


39 

3-9 
7.8 
11.7 
15-6 
19-5 
23-4 
27-3 


36.0I35.1 


14.4 
18.0 
21.6 


25-9  25-2 
29.6  28.8 
34-2  33-3  32-4 


33 

3-3 
6.6 
9.9 
13.2 
16.5 
19.8 
23.1 
26.4 
29.7 


35 

34 

3-5 
7.0 

3-4 
6.8 

io.,s 

10.2 

14.0 

136 

17-5 

17.0 

21.0 

20.4 

24-5 
28.0 

23.« 
27.2 

31-5 

30.6 1 

31  30 


3-0 
6.0 
9.0 
12.0 
15.0 
18.0 
21.0 


24.8  24.0 
27.9I27.0 


P.P. 


105 


N. 

L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

150 

151 
152 
153 

17  609 

638 

667 

696 

725 

754 

782 

*070 

355 

639 

811 

*099 

384 

667 

840 

*127 

412 

696 

869 

*156 

441 

724 

1^  1898 

18/184 

469 

926 
213 
498 

955 
241 
526 

984 
270 
554 

*013 
298 

583 

*041 
327 
611 

I 

2 

29 

2.9 

5.8 

28 

2.8 
5.6 

154 
155 
156 

i^}752 

^^  ^  033 

312 

780 
061 
340 

808 
089 
368 

837 
117 
396 

865 
145 
424 

893 
173 
451 

921 
201 
479 

949 

229 
507 

977 

257 
535 

*005 
285 
562 

3 

4 
5 
6 

8.7 
11.6 
14-5 
17.4 

8.4 
II. 2 
14.0 
16.8 

157 
158 
159 
160 

19  1  ^^^ 
^^866 

^"  J  140 
412 

618 
893 
167 
439 

645 
921 
194 

673 

948 
222 

700 
976 
249 

520 

728 

*003 

276 

548 

756 

*030 

303 

575 

783 

*058 

330 

602 

811 

*085 

358 

629 

838 

*112 

385 

656 

7 
8 
9 

20.3 
23.2 
26.1 

22.4 
25.2 

466 

493 

1 

161 
162 
163 

2^^219 

710 
978 

245 

737 
*005 

272 

763 

*032 

299 

•790 

*059 

325 

817 

*085 

352 

844 

*112 

378 

871 

*139 

405 

898 

*165 

431 

925 

*192 

458 

I 

2 

27 

2.7 

5-4 

26 

2.6 

5-2 

164 
165 
166 

484 

748 

22  Oil 

511 

775 
037 

537 
801 
063 

564 
827 
089 

590 
854 
115 

617 
880 
141 

643 
906 
167 

669 
932 
194 

696 
958 
220 

722 
985 
246 

3 
4 
5 
6 

8.1 
10.8 
13.S 
16.2 

7.8 

10.4 

13.0 

15.6 

167 
168 
169 
170 

171 
172 
173 

272 
22.531 

045 

300 

23.553 

24}  805 

%55 

304 

it]"' 

285 

527 

298 
557 
814 
070 

324 
583 
840 
096 

350 
608 
866 
121 
376 
629 
880 

376 
634 
891 
147 
401 
654 
905 

401 
660 
917 

427 
686 
943 

453 
712 
968 

479 
737 
994 

505 

763 

*019 

i 

9 

iS.y 
21.6 
24.3 

20.8 
23./ 

172 
426 
679 
930 

198 
452 
704 
955 

223 
477 
729 
980 

249 

502 

754 

*005 

274 

528 

779 

*030 

325 
578 
830 

350 
603 

855 

25 

1  2.S 

2  5.0 

174 
175 
176 

080 
329 
576 

105 
353 
601 

130 
378 
625 

155 
403 
650 

180 
428 
674 

204 
452 
699 

229 

477 
724 

254 
502 
748 

279 

527 
773 

3  7.5 

4  1 0.0 

5  12.5 

6  is.o 

177 
178 
179 

822 
066 
310 
551 

846 
091 

334 

575 

871 
115 
358 
600 

895 
139 
382 
624 

920 
164 
406 

944 

188 
431 

969 

212 

455 

993 
237 
479 

*018 
261 
503 

717.S 

8  20.0 

9  22.5 

180 

648 

672 

696 

720 

744 

181 
182 
183 

768 

26  007 

245 

792 
031 
269 

816 
055 
293 

840 
079 
316 

864 
102 
340 

888 
126 
364 

912 
150 

387 

935 
174 
411 

959 
198 
435 

983 
221 

458 

I 

2 

24 

2.4 
4.8 
7.2 
9.6 
I2.C 
14.4 

23 

2.3 

4  6 

184 
185 
186 

482 

'Hill 

416 

330 

556 

505 
741 
975 

529 
764 
998 

553 

788 

*021 

576 

811 

*045 

600 

834 

*068 

623 

858 

*091 

647 

881 

*114 

670 

905 

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4.2 

4.8 

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767 
768 
769 

480 
536 
593 

485 
542 
598 

491 
547 
604 

497 
553 
610 

502 
559 
615 

508 
564 
621 

513 
570 
627 

519 
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632 

525 
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530 
587 
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770 

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689 

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700 
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812 
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771 

772 
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705 
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098 
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104 
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109 
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115 
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120 
176 

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126 

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076 
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187 

081 
137 
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087 
143 
198 

092 
148 
204 

780 

215 

221 

226 

232 
287 
343 
398 

237 
293 
348 
404 

243 
298 
354 
409 

248 
304 
360 

415 

254 

260 

781 
782 
783 

265 
321 
376 

271 
326 

382 

276 
332 

387 

282 
337 
393 

310 
365 
421 

315 
371 
426 

784 
785 
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432 
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542 

437 
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548 

443 
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553 

448 
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476 
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653 
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609 
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625 
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793 

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840 
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856 
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916 
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922 
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794 
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988 
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119 

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146 
200 

255 

151 
206 
260 

157 
211 
266 

162 
217 
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168 
222 
276 

173 

227 
282 

179 

233 
287 

184 
238 
293 

189 
244 
298 

195 
249 
304 

358 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

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1 

2 

3 

4 

5 

6 

7 

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1 

2 

3 

4 

5 

6 

7 

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800 

90  309 

314 

320 

325 

331 

336 

342 

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352 

358 
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801 
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363 
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369 
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374 

428 
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380 

434 
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390 
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396 
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804 
805 
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634 

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536 
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542 
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547 
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612 
666 

563 
617 
671 

569 
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807 
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687 
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693 
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800 

698 

752 
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757 
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709 
763 
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714 
768 
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773 
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169 

068 
121 
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073 
126 
180 

078 
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100 
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105 
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110 
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817 
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222 
275 
328 

228 
281 
334 

233 
286 
339 

238 
291 
344 

243 
297 
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249 
302 
355 
408 
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514 
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254 
307 
360 

259 
312 
365 
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265 
318 
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270 

323 
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440 
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498 
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466 
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524 
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645 
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598 
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603 
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614 
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630 
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735 

635 
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772 
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832 
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117 
169 
221 

122 
174 
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127 
179 
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132 
184 
236 

137 
189 
241 

143 
195 

247 

148 
200 
252 

153 
205 
257 

158 
210 
262 

163 
215 
267 

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2 
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1.5 

837 
838 
839 

273 
324 
376 

278 
330 
381 

283 
335 
387 

288 
340 
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293 
345 
397 

298 
350 
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304 
355 
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309 
361 
412 

314 
366 
418 

319 
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840 

428 
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433 

438 

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547 
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449 
500 

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459 

464 
516 
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490 
542 
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634 
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737 

639 
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742 

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716 
768 

670 

722 
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675 

727 
778 

681 

732 
783 

847 
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793 
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850 
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809 
860 
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875 
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932 

834 
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937 

850 

942 

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957 

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967 

973 

978 

983 

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L.   0 

1 

2 

3 

4 

5 

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N. 

L.   0 

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2 

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6 

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105 

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110 

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855 
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146 
197 
247 

151 
202 
252 

156 
207 

258 

161 
212 
263 

166 
217 
268 

171 
222 

273 

176 

227 
278 

181 
232 
283 

186 
237 
288 

192 
242 
293 

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2 

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858 
859 

298 
349 
399 
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303 
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404 
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308 
359 
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313 
364 
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318 
369 
420 

323 
374 
425 

328 
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334 
384 
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339 
389 
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344 
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546 
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470 

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626 

480 

485 
536 
586 
636 

861 
862 
863 

500 
551 
601 

505 
556 
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520 
571 
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864 
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651 
702 

752 

656 
707 

757 

661 
712 
762 

666 
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767 

671 

722 
772 

676 

727 
777 

682 
732 
782 

687 
737 
787 

692 
742 
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697 
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867 
868 
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802 
852 
902 
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807 
857 
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812 
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912 

817 
867 
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822 
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922 

827 
877 
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832 
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932 
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937 

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892 
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957 

962 

967 

972 

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052 
101 

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057 
106 

012 
062 
111 

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067 
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022 
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126 

032 
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131 

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141 

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874 
875 
876 

151 
201 
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156 
206 

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161 
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260 

166 
216 
265 

171 
221 
270 

176 
226 

275 

181 
231 
280 

186 
236 
285 

191 
240 
290 

196 

245 
295 

3 
4 
5 
6 
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2.0 
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35 
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877 
878 
879 
880 

300 
349 
399 

305 
354 
404 

310 
359 
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458 

315 
364 
414 

320 
369 
419 

325 
374 
424 
473 

330 
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429 

335 
384 
433 
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340 
389 
438 
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537 
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635 

345 
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542 
591 
640 

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453 

463 
512 
562 
611 

468 

478 

881 
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883 

498 
547 
596 

503 
552 
601 

507 
557 
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517 
567 
616 

522 
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527 
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532 
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645 
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699 

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655 
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753 

660 
709 

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714 
763 

670 
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768 

675 

724 
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680 
729 

778 

685 
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689 

738 
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134 
182 
231 

139 
187 
236 

143 
192 
240 

148 
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153 
202 
250 

158 
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163 
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168 
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265 

173 
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177 
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279 
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284 
332 
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289 
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294 
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390 
439 

299 

347 
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352 
400 
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308 
357 
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313 
361 
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318 
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415 
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323 
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L.   0 

1 

2 

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900 

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521 
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530 
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535 
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617 
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622 
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626 
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722 

631 
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727 

636 

684 
732 

641 
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646 
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751 

660 
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256 
303 
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261 
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961 
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272 
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556 
601 
646 

561 
605 
650 

565 
610 
655 

570 
614 
659 

574 
619 
664 

579 
623 
668 

583 
628 
673 

970 

677 

682 

686 

691 

695 

700 

704 
749 
793 
838 

709 
753 
798 
843 

713 

717 

971 
972 
973 

722 
767 
811 

726 
771 
816 

731 
776 
820 

735 
780 
825 

740 
784 
829 

744 
789 
834 

758 
802 
847 

762 
807 
851 

974 
975 
976 

856 

900 

..  945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 
981 

896 
941 
985 

977 
978 
979 

'^  ^  034 
078 

994 

038 
083 

998 
043 

087 

*003 
047 
092 

*007 
052 
096 

*012 
056 
100 

*016 
061 
105 
149 
193 
238 
282 

*021 
065 
109 
154 
198 
242 
286 

*025 
069 
114 
158 
202 
247 
291 

*029 
074 
118 
162 
207 
251 
295 

980 

123 

127 

131 

136 

140 

145 

981 
982 
983 

167 
211 

255 

171 
216 
260 

176 
220 
264 

180 
224 
269 

185 
229 

273 

189 
233 

277 

984 
985 
986 

300 
344 

388 

304 
348 
392 

308 
352 
396 

313 
357 
401 

317 
361 
405 

322 
366 
410 

326 
370 
414 

330 
374 
419 

335 
379 
423 

339 

3S3 
427 

I 

2 

4 

987 
988 
989 

432 
476 
520 

436 

480 
524 

441 

484 
528 
572 

445 
489 
533 

449 
493 

537 

454 
498 

542 

585 

458 
502 
546 
590 

463 
506 
550 
594 

467 
511 

555 

471 
515 
559 

3 

4 

1 

7 
8 
9 

1.2 

1.6 

2.0 
2.4 
2.8 
3.2 
3.6 

990 

564  I   568 

577 

581 

599 

603 

991 
992 
993 

607 
651 
695 

612 
656 
699 

616 
660 
704 

621 
664 

708 

625 
669 
712 

629 
673 
717 

634 
677 
721 

638 
682 
726 

642 
686 
730 

647 
691 

734 

994 
995 
996 

739 

782 
826 

743 
787 
830 

747 
791 
835 

752 
795 
839 

756 
800 

843 

760 
804 

848 

765 
808 
852 

769 
813 
856 

774 
817 
861 

778 
822 
865 

997 
998 
999 

870 
913 
957 

874 
917 
961 

878 
922 
965 

883 
926 
970 

887 
930 
974 

891 
935 
978 

896 
939 
983 

900 
944 
987 

904 
948 
991 

035 

909 
952 
996 
039 

1000 

00  000  004 

009 

013 

017 

022 

026 

030 

N. 

L.   0  1  1 

2 

3 

4 

5  1  6 

7 

8 

9 

P.P. 

TABLE  II, 


LOGARITHMS 

OF   THE 

TRIGONOMETRIC   FUNCTIONS. 


123 


124 

0° 

S. 

—  4. 

557 

T. 

It 

/ 

L.  Sin. 

L.  Tan. 

L.  Cot. 

L.  Cos. 

60 

59 

58 

57 

557 

0 

0 

0.00000 

557 
557 
557 

557 

557 
557 

60 
120 
180 

1 

2 
3 

6.46373 
6.76476 
6.94085 

6.46373 
6.76476 
6.94085 

3.53627 
3.23524 
3.05915 

0.00000 
0.00000 
0.00000 

557 
557 
557 

558 
558 
558 

240 
300 
360 

4 
5 
6 

7.06579 
7.16270 
7.24188 

7.06579 
7.16270 
7.24188 

2.93421 
2.83730 
2.75812 

0.00000 
0.00000 
0.00000 

56 

55 
54 

557 
557 
557 

558 
5S8 
558 

420 
480 
540 

7 
8 
9 

7.30882 
7.36682 
7.41797 

7.30882 
7.36682 
7.41797 

2.69118 
2.63318 
2.58203 

0.00000 
0.00000 
0.00000 

53 
52 
51 
50 

49 

48 
47 

557 

558 

600 

10 

7.46373 

7.46373 

2.53627 

0.00000 

557 
557 
557 

558 
558 
558 

660 
720 
780 

11 
12 
13 

7.50512 
7.54291 

7.57767 

7.50512 
7.54291 

7.57767 

2.49488 
2.45709 
2.42233 

0.00000 
0.00000 
0.00000 

557 
557 
557 

558 
358 
558 

840 
900 
960 

14 
15 
16 

7.60985 
7.63982 
7.66784 

7.60986 
7.63982 
7.66785 

2.39014 
2.36018 
2.33215 

0.00000 
0.00000 
0.00000 

46 
45 
44 

557 
557 
557 

558 
558 
558 

1020 
1080 
1 140 

17 
18 
19 

7.69417 
7.71900 

7.74248 

7.69418 
7.71900 

7.74248 

2.30582 
2.28100 
2.25752 

9.99999 
9.99999 
9.99999 

43 
42 
41 
40 

39 
38 
37 

557 

558 

I2CX) 

20 

7.76475 

7.76476 

2.23524 

9.99999 

557 
557 
557 

558 
558 
558 

1260 
1320 
1380 

21 
22 
23 

7.78594 
7.80615 
7.82545 

7.78595 
7.80615 
7.82546 

2.21405 
2.19385 
2.17454 

9.99999 
9.99999 
9.99999 

557 
557 
557 

558 

558 
558 

1440 
1500 
1560 

24 
25 
26 

7.84393 
7.86166 
7.87870 

7.84394 
7.86167 

7.87871 

2.15606 
2.13833 
2.12129 

9.99999 
9.99999 
9.99999 

36 

35 
34 

557 
557 
557 

558 
558 
559 

1620 
1680 
1740 

27 
28 
29 

7.89509 
7.91088 
7.92612 

7.89510 
7.91089 
7.92613 

2.10490 
2.08911 
2.07387 

9.99999 
9.99999 
9.99998 

2>^ 
32 
31 
30 

557 

559 

1800 

30 

7.94084 

7.94086 

2.05914 

9.99998 

557 
557 
557 

559 
559 
559 

i860 
1920 
1980 

31 
32 
33 

7.95508 
7.96887 
7.98223 

7.95510 
7.96889 

7.98225 

2.04490 
2.03111 
2.01775 

9.99998 
9.99998 
9.99998 

29 
28 

27 

557 
557 
557 

559 
559 
559 

2040 
2100 
2160 

34 
35 
36 

7.99520 
8.00779 
8.02002 

7.99522 
8.00781 
8.02004 

2.00478 
1.99219 
1.97996 

9.99998 
9.99998 
9.99998 

26 

25 
24 

557 
557 
557 

559 
559 
559 

2220 
2280 
2340 

3^ 
38 
39 

8.03192 
8.04350 
8.05478 

8.03194 
8.04353 
8.05481 

1.96806 
1.95647 
1.94519 

9.99997 
9.99997 
9.99997 

23 
22 
21 
20 

557 

559 

2400 

40 

8.06578 

8.06581 

1.93419 

9.99997 

556 
556 
556 

560 
560 
560 

2460 
2520 
2580 

41 
42 
43 

8.07650 
8.08696 
8.09718 

8.07653 
8.08700 
8.09722 

1.92347 
1.91300 
1.90278 

9.99997 
9.99997 
9.99997 

19 
18 
17 

556 
556 
556 

560 
560 
560 

2640 
2700 
2760 

44 
45 
46 

8.10717 
8.11693 
8.12647 

8.10720 
8.11696 
8.f2651 

1.89280 
1.88304 
1.87349 

9.99996 
9.99996 
9.99996 

16 
15 
14 

556 
556 
556 

560 
560 

560 

2820 
2880 
2940 

47 
48 
49 

8.13581 
8.14495 
8.15391 

8.13585 
8.14500 
8.15395 

1.86415 
1.85500 
1.84605 

9.99996 
9.99996 
9.99996 

13 
12 
11 

556 

S6i 

3000 

50 

8.16268 

8.16273 

1.83727 

9.99995 

10 

9 

8 
7 

556 
556 
556 

561 

561 
561 

3060 
3120 
3180 

51 
52 
53 

8.17128 
8.17971 
8.18798 

8.17133 
8.17976 
8.18804 

1.82867 
1.82024 
1.81196 

9.99995 
9.99995 
9.99995 

556 
556 
556 

561 
561 

3240 
3300 
3360 

54 
55 
56 

8.19610 
8.20407 
8.21189 

8.19616 
8.20413 
8.21195 

1.80384 
1.79587 
1.78805 

9.99995 
9.99994 
9.99994 

6 

5 
4 

555 
555 
555 

561 
562 

562 

3420 
3480 
3540 

57 
58 
59 

8.21958 
8.22713 
8.23456 

8.21964 
8.22720 
8.23462 

1.78036 
1.77280 
1.76538 

9.99994 
9.99994 
9.99994 

3 
2 
1 

555 

562 

3600 

60 

8.24186 

8.24192 

1.75808 

9.99993 

0 

1 

L.  Cos. 

L.  Cot. 

L.  Tan. 

L.  Sin. 

V 

125 

s. 

4. 

555 

T. 

n 

t 

L.  Sin. 

L.  Tan. 

L.  Cot. 

L.  Cos. 

562 

3600 

0 

8.24186 

8.24192 

1.75808 

9.99993 

60 

59 

58 
57 

555 
555 
555 

562 
562 

562 

3660 
3720 
3780 

1 

2 
3 

8.24903 
8.25609 
8.26304 

8.24910 
8.25616 
8.26312 

1.75090 
1.74384 
1.73688 

9.99993 
9.99993 
9.99993 

555 
555 
555 

563 
563 
563 

3840 
3900 
3960 

4 
5 
6 

8.26988 
8.27661 
8.28324 

8.26996 
8.27669 
8.28332 

1.73004 
1.72331 
1.71668 

9.99992 
9.99992 
9.99992 

56 

55 
54 

555 
555 
555 

563 
563 
563 

4020 
4080 
4140 

7 
8 
9 

8.28977 
8.29621 
8.30255 

8.28986 
8.29629 
8.30263 

1.71014 
1.70371 
1.69737 

9.99992 
9.99992 
9.99991 

53 
52 
51 
50 

554 

563 

4200 

10 

8.30879 

8.30888 

1.69112 

9.99991 

554 
554 
554 

564 
564 
564 

4260 
4320 
4380 

11 
12 
13 

8.31495 
8.32103 
8.32702 

8.31505 
8.32112 
8.32711 

1.68495 
1.67888 
1.67289 

9.99991 
9.99990 
9.99990 

49 

48 
47 

554 
554 
554 

564 
564 
565 

4440 
4500 
4560 

14 
15 
16 

8.33292 
8.33875 
8.34450 

8.33302 
8.33886 
8.34461 

1.66698 
1.66114 
1.65539 

9.99990 
9.99990 
9.99989 

46 
45 
44 

554 
554 
554 

565 
565 
56s 

4620 
4680 
4740 

17 
18 
19 

8.35018 
8.35578 
8.36131 

8.35029 
8.35590 
8.36143 

1.64971 
1.64410 
1.63857 

9.99989 
9.99989 
9.99989 

43 
42 
41 

554 

56s 

4800 

20 

8.36678 

8.36689 

1.63311 

9.99988 

40 

553 
553 
553 

566 
566 
566 

4860 
4920 
4980 

21 
22 

23 

8.37217 
8.37750 
8.38276 

8.37229 
8.37762 
8.38289 

1.62771 
1.62238 
1.61711 

9.99988 
9.99988 
9.99987 

39 

38 
37 

553 
553 
553 

566 
566 
567 

5040 
5100 
5160 

24 
25 
26 

8.38796 
8.39310 
8.39818 

8.38809 
8.39323 
8.39832 

1.61191 
1.60677 
1.60168 

9.99987 
9.99987 
9.99986 

36 
35 
34 

553 
553 
553 

567 
567 
567 

5220 
5280 
5340 

27 
28 
29 

8.40320 
8.40816 
8.41307 

8.40334 
8.40830 
8.41321 

1.59666 
1.59170 
1.58679 

9.99986 
9.99986 
9.99985 

33 
32 
31 
30 

553 

567 

5400 

30 

8.41792 

8.41807 

1.58193 

9.99985 

552 
552 
552 

568 
568 
568 

5460 
5520 
5580 

31 

32 

8.42272 
8.42746 
8.43216 

8.42287 
8.42762 
8.43232 

1.57713 
1.57238 
1.56768 

9.99985 
9.99984 
9.99984 

29 

28 
27 

552 
552 
552 

568 
569 

569 

5640 
5700 
5760 

34 
35 
36 

8.43680 
8.44139 
8.44594 

8.43696 
8.44156 
8.44611 

1.56304 
1.55844 
1.55389 

9.99984 
9.99983 
9.99983 

26 
25 
24 

552 

552 
551 

569 
569 
569 

5820 
5880 
5940 

37 
38 
39 

8.45044 
8.45489 
8.45930 

8.45061 
8.45507 
8.45948 

1.54939 
1.54493 
1.54052 

9.99983 
9.99982 
9.99982 

23 
22 
21 

551 

570 

6000 

40 

8.46366 

8.46385 

1.53615 

9.99982 

20 

19 
18 
17 

551 
551 
551 

570 
570 
570 

6060 
6120 
6180 

41 
42 
43 

8.46799 
8.47226 
8.47650 

8.46817 
8.47245 
8.47669 

1.53183 
1.52755 
1.52331 

9.99981 
9.99981 
9.99981 

551 
551 
551 

571 
571 
571 

6240 
6300 
6360 

44 

45 
46 

8.48069 
8.48485 
8.48896 

8.48089 
8.48505 
8.48917 

1.51911 
1.51495 
1.51083 

9.99980 
9.99980 
9.99979 

16 
15 
14 

550 
550 
550 

572 
572 
572 

6420 
6480 
6540 

47 
48 
49 

8.49304 
8.49708 
8.50108 

8.49325 
8.49729 
8.50130 

1.50675 
1.50271 
1.49870 

9.99979 
9.99979 
9.99978 

13 
12 
11 
10 

9 

8 

7 

'     550 

572 

6600 

50 

8.50504 

8.50527 

1.49473 

9.99978 

550 
550 
550 

573 
573 
573 

6660 
6720 
6780 

51 

52 
53 

8.50897 
8.51287 
8.51673 

8.50920 
8.51310 
8.51696 

1.49080 
1.48690 
1.48304 

9.99977 
9.99977 
9.99977 

550 
549 
549 

573 
574 
574 

6840 
6900 
6960 

^4 
55 
56 

8.52055 
8.52434 
8.52810 

8.52079 
8.52459 
8.52835 

1.47921 
1.47541 
1.47165 

9.99976 
9.99976 
9.99975 

6 

5 
4 

549 
549 
549 

574 

575 
575 

7020 
7080 
7140 

57 
58 
59 

8.53183 
8.53552 
8.53919 

8.53208 
8.53578 
8.53945 

1.46792 

1.46422 
1.46055 

9.99975 
9.99974 
9.99974 

3 
2 
1 
0 

549 

575 

7200 

60 

8.54282 

8.54308 

1.45692 

9.99974 

L.  Cos. 

L.  Cot. 

L.  Tan. 

L.  Sin. 

/ 

126 

2° 

s. 

549 

T. 

68—   >.• 

n 

t 

L.  Sin. 

L.  Tan. 

L.  Cot. 

L.  Cos. 

575 

7200 

0 

8.54282 

8.54308 

1.45692 

9.99974 

60 

549 
548 
548 

575 
576 
576 

7260 
7320 
7380 

1 

2 
3 

8.54642 
8.54999 
8.55354 

8.54669 
8.55027 
8.55382 

1.45331 
1.44973 
1.44618 

9.99973 
9.99973 
9.99972 

59 

58 
57 

548 
548 
548 

576 
577 
577 

7440 
7500 
7560 

4 
5 
6 

8.55705 
8.56054 
8.56400 

8.55734 
8.56083 
8.56429 

1.44266 
1.43917 
1.43571 

9.99972 
9.99971 
9.99971 

56 

55 
54 

548 
547 
547 

577 
578 
578 

7620 
7680 
7740 

7 
8 
9 

8.56743 
8.57084 
8.57421 

8.56773 
8.57114 
8.57452 

1.43227 
1.42886 
1.42548 

9.99970 
9.99970 
9.99969 

53 
52 
51 

547 

578 

579 
579 
579 

7800 

10 

8.57757 

8.57788 

1.42212 

9.99969 

50 

547 
547 
547 

7860 
7920 
7980 

11 
12 
13 

8.58089 
8.58419 

8.58747 

8.58121 
8.58451 
8.58779 

1.41879 
1.41549 
1.41221 

9.99968 
9.99968 
9.99967 

49 
48 
47 

546 
546 
546 

579 
580 
580 

8040 
8100 
8160 

14 
15 
16 

8.59072 
8.59395 
8.59715 

8.59105 
8.59428 
8.59749 

1.40895 
1.40572 
1.40251 

9.99967 
9.99967 
9.99966 

46 
45 
44 

546 
546 
546 

580 
581 
581 

8220 
8280 
8340 

17 
18 
19 

8.60033 
8.60349 
8.60662 

8.60068 
8.60384 
8.60698 

1.39932 
1.39616 
1.39302 

9.99966 
9.99965 
9.99964 

43 
42 
41 

545 

582 

8400 

20 

8.60973 

8.61009 

1.38991 

9.99964 

40 

39 

38 
37 

545 
545 
545 

582 
582 
583 

8460 
8520 
8580 

21 
22 
23 

8.61282 
8.61589 
8.61894 

8.61319 
8.61626 
8.61931 

1.38681 
1.38374 
1.38069 

9.99963 
9.99963 
9.99962 

545 
545 
544 

583 
583 
584 

8640 
8700 
8760 

24 
25 
26 

8.62196 
8.62497 
8.62795 

8.62234 
8.62535 
8.62834 

1.37766 
1.37465 
1.37166 

9.99962 
9.99961 
9.99961 

36 
35 
34 

544 
544 

544 

584 
584 
585 

8820 
8880 
8940 

27 
28 
29 

8.63091 
8.63385 
8.63678 

8.63131 
8.63426 
8.63718 

1.36869 
1.36574 
1.36282 

9.99960 
9.99960 
9.99959 

32 
31 

544 

544 
543 
543 

585 

9000 

30 

8.63968 

8.64009 

1.35991 

9.99959 

30 

29 

28 
27 

585 
586 
586 

9060 
9120 
9180 

31 
32 
33 

8.64256 
8.64543 
8.64827 

8.64298 
8.64585 
8.64870 

1.35702 
1.35415 
1.35130 

9.99958 
9.99958 
9.99957 

543 
543 
543 

587 
587 
587 

9240 
9300 
9360 

34 
35 
36 

8.65110 
8.65391 
8.65670 

8.65154 
8.65435 
8.65715 

1.34846 
1.34565 
1.34285 

9.99956 
9.99956 
9.99955 

26 
25 
24 

542 
542 
542 

588 
588 
588 

9420 
9480 
9540 

37 
38 
39 

8.65947 
8.66223 
8.66497 

8.65993 
8.66269 
8.66543 

1.34007 
1.33731 
1.33457 

9.99955 
9.99954 
9.99954 

23 
22 
21 

542 

589 

9600 

40 

8.66769 

8.66816 

1.33184 

9.99953 

20 

542 
541 
541 

589 
590 
590 

9660 
9720 
9780 

41 
42 
43 

8.67039 
8.67308 
8.67575 

8.67087 
8.67356 
8.67624 

1.32913 
1.32644 
1.32376 

9.99952 
9.99952 
9.99951 

19 
18 
17 

541 
541 
541 

590 
591 
591 

9840 
9900 
9960 

44 
45 
46 

8.67841 
8.68104 
8.68367 

8.67890 
8.68154 
8.68417 

1.32110 
1.31846 
1.31583 

9.99951 
9.99950 
9.99949 

16 
15 
14 

540 
540 
540 
540 
540 
539 
539 

592 
592 
592 

593 

10020 
10080 
10140 

47 
48 
49 

8.68627 
8.68886 
8.69144 

8.68678 
8.68938 
8.69196 

1.31322 
1.31062 
1.30804 

9.99949 
9.99948 
9.99948 

13 
12 
11 

10200 

50 

8.69400 

8.69453 

1.30547 

9.99947 

10 

593 
594 
594 

10260 
10320 
10380 

51 
52 
53 

8.69654 
8.69907 
8.70159 

8.69708 
8.69962 
8.70214 

1.30292 
1.30038 
1.29786 

9.99946 
9.99946 
9.99945 

9 

8 

7 

539 
539 
539 

595 

595 
595 

10440 
10500 
10560 

54 
55 
56 

8.70409 
8.70658 
8.70905 

8.70465 
8.70714 
8.70962 

1.29535 
1.29286 
1.29038 

9.99944 
9.99944 
9.99943 

6 

5 
4 

538 
538 
538 

596 
596 
597 

10620 
10680 
10740 

57 
58 
59 

8.71151 
8.71395 
8.71638 

8.71208 
8.71453 
8.71697 

1.28792 

1.28547 
1.28303 

9.99942 
9.99942 
9.99941 

3 
2 
1 
0 

538 

597 

loSoo 

60 

8.71880 

8.71940 

1.28060 

9.99940 

L.  Cos. 

L.  Cot. 

L.  Tan. 

L.  Sin. 

1 

3° 

127 

s. 

538 

T. 

II 

f 

L.  Sin. 

L.  Tan. 

L.  Cot. 

L.  Cos. 

597 

10800 

0 

8.71880 

8.71940 

1.28060 

9.99940 

60 

537 
537 
537 

598 
598 
599 

10860 
ioy20 
10980 

1 

2 
3 

8.72120 
8.72359 
8.72597 

8.72181 
8.72420 
8.72659 

1.27819 
1.27580 
1.27341 

9.99940 
9.99939 
9.99938 

59 

58 

57 

537 
537 
536 

599 
599 
600 

1 1040 

IIIOO 

11160 

4 
5 
6 

8.72834 
8.73069 
8.73303 

8.72896 
8.73132 
8.73366 

1.27104 
1.26868 
1.26634 

9.99938 
9.99937 
9.99936 

56 
55 
54 

536 
536 
536 

600 
601 
6oi 

11220 
11280 
11340 

7 
8 
9 

8.73535 
8.73767 
8.73997 

8.73600 
8.73832 
8.74063 

1.26400 
1.26168 
1.25937 

9.99936 
9.99935 
9.99934 

53 
52 
51 

535 

602 

11400 

10 

8.74226 

8.74292 

1.25708 

9.99934 

50 

535 
535 
535 

602 
603 
603 

1 1460 
1 1 520 
11580 

11 
12 
13 

8.74454 
8.74680 
8.74906 

8.74521 
8.74748 
8.74974 

1.25479 

1.25252 
1.25026 

9.99933 
9.99932 
9.99932 

49 

48 
47 

534 
534 
534 

604 

604 
605 

1 1 640 
II 700 
II 760 

14 
15 
16 

8.75130 
8.75353 
8.75575 

8.75199 
8.75423 
8.75645 

1.24801 
1.24577 
1.24355 

9.99931 
9.99930 
9.99929 

46 

45 
44 

534 
533 
533 

605 
606 
606 

11820 
11880 
11940 

17 
18 
19 

8.75795 
8.76015 
8.76234 

8.75867 
8.76087 
8.76306 

1.24133 
1.23913 
1.23694 

9.99929 
9.99928 
9.99927 

43 
42 
41 

533 

607 

12000 

20 

8.76451 

8.76525 

1.23475 

9.99926 

40 

39 

38 
37 

533 
532 
532 

607 
608 
608 

12060 
12120 
12180 

21 
22 
23 

8.76667 
8.76883 
8.77097 

8.76742 
8.76958 
8.77173 

1.23258 
1.23042 
1.22827 

9.99926 
9.99925 
9.99924 

532 
532 
531 

609 
609 
610 

12240 
12300 
12360 

24 
25  . 
26 

8.77310 

8.77522 
8.77733 

8.77387 
8.77600 
8.77811 

1.22613 
1.22400 
1.22189 

9.99923 
9.99923 
9.99922 

36 
35 
34 

531 
531 
531 

610 
6ii 
611 

12420 
12480 
12540 

27 
28 
29 

8.77943 
8.78152 
8.78360 

8.78022 
8.78232 
8.78441 

1.21978 
1.21768 
1.21559 

9.99921 
9.99920 
9.99920 

33 
32 
31 

530 

612 

12600 

30 

8.78568 

8.78649 

1.21351 

9.99919 

30 

530 
530 
530 

612 
613 
613 

12660 
12720 
12780 

31 

32 

8.78774 
8.78979 
8.79183 

8.78855 
8.79061 
8.79266 

1.21145 
1.20939 
1.20734 

9.99918 
9.99917 
9.99917 

29 
28 
27 

529 
529 
529 

614 
614 
61S 

12840 
12900 
12960 

34 
35 
36 

8.79386 
8.79588 
8.79789 

8.79470 
8.79673 
8.79875 

1.20530 
1.20327 
1.20125 

9.99916 
9.99915 
9.99914 

26 
25 
24 

529 
528 
528 

615 
616 
616 

13020 
13080 
13140 

37 
38 
39 

8.79990 
8.80189 
8.80388 

8.80076 
8.80277 
8.80476 

1.19924 
1.19723 
1.19524 

9.99913 
9.99913 
9.99912 

23 
22 
21 

528 

617 

13200 

40 

8.80585 

8.80674 

1.19326 

9.99911 

20 

528 
527 
527 

617 
618 
618 

13260 
13320 
13380 

41 
42 
43 

8.80782 
8.80978 
8.81173 

8.80872 
8.81068 
8.81264 

1.19128 
1.18932 
1.18736 

9.99910 
9.99909 
9.99909 

19 
18 
17 

527 
526 
526 

619 
620 
620 

13440 
13500 
13560 

44 
45 
46 

8.81367 
8.81560 
8.81752 

8.81459 
8.81653 
8.81846 

1.18541 
1.18347 
1.18154 

9.99908 
9.99907 
9.99906 

16 
15 
14 

526 
526 

525 

621 
621 
622 

13620 
13680 
13740 

47 
48 
49 

8.81944 
8.82134 
8.82324 

8.82038 
8.82230 
8.82420 

1.17962 
1.17770 
1.17580 

9.99905 
9.99904 
9.99904 

13 
12 
11 

525 

622 

13800 

50 

8.82513 

8.82610 

1.17390 

9.99903 

10 

525 
525 
524 

623 
623 
624 

13860 
13920 
13980 

51 
52 
53 

8.82701 
8.82888 
8.83075 

8.82799 
8.82987 
8.83175 

1.17201 
1.17013 
1.16825 

9.99902 
9.99901 
9.99900 

9 

8 

7 

524 
524 
523 

62s 
625 
626 

14040 
14100 
14160 

54 
55 
56 

8.83261 
8.83446 
8.83630 

8.83361 
8.83547 
8.83732 

1.16639 
1.16453 
1.16268 

9.99899 
9.99898 
9.99898 

6 
5 
4 

523 
523 

522 

626 
627 
628 

14220 
14280 
14340 

57 
58 
59 

8.83813 
8.83996 
8.84177 

8.83916 
8.84100 
8.84282 

1.16084 
1.15900 
1.15718 

9.99897 
9.99896 
9.99895 

3 
2 

1 

522 

628 

14400 

60 

8.84358 

8.84464 

1.15536 

9.99894 

0 

L.  Cos. 

L.  Cot. 

L.Tan. 

L.  Sin. 

/ 

l^O 

* 

S. 

4. 

522 

T. 

It 

/ 

L.  Sin. 

L.  Tan. 

L.  Cot. 

L.  Cos. 

628 

14400 

0 

8.84358 

8.84464 

1.15536 

9.99894 

60 

522 
522 
521 

629 
629 

630 

14460 
14520 
14580 

1 

2 
3 

8.84539 
8.84718 
8.84897 

8.84646 
8.84826 
8.85006 

1.15354 
1.15174 
1.14994 

9.99893 
9.99892 
9.99891  - 

59 

58 
57 

521 
521 
520 

631 
631 
632 

14640 
14700 
14760 

4 
5 
6 

8.85075 
8.85252 
8.85429 

8.85185 
8.85363 
8.85540 

1.14815 
1.14637 
1.14460 

9.99891 
9.99890 
9.99889 

56 
55 
54 

520 
520 
520 

632 
633 
634 

14820 
14880 
14940 

7 
8 
9 

8.85605 
8.85780 
8.85955 

8.85717 
8.85893 
8.86069 

1.14283 
1.14107 
1.13931 

9.99888 
9.99887 
9.99886 

53 
52 
51 

519 

634 

15000 

10 

8.86128 

8.86243 

1.13757 

9.99885 

50 

519 
519 
518 

635 
635 
636 

15060 
15120 
15180 

11 
12 
13 

8.86301 
8.86474 
8.86645 

8.86417 
8.86591 
8.86763 

1.13583 
1.13409 
1.13237 

9.99884 
9.99883 
9.99882 

49 

48 

47 

518 
518 
517 

637 
637 

638 

15240 
15300 
15360 

14 
15 
16 

8.86816 
8.86987 
8.87156 

8.86935 
8.87106 

8.87277 

1.13065 
1.12894 
1.12723 

9.99881 
9.99880 
9.99879 

46 

45 
44 

517 
517 
516 

638 
639 

640 

15420 
15480 
15540 

17 
18 
19 

8.87325 
8.87494 
8.87661 

8.87447 
8.87616 

8.87785 

1.12553 
1.12384 
1.12215 

9.99879 
9.99878 
9.99877 

43 
42 
41 

516 

640 

15600 

20 

8.87829 

8.87953 

1.12047 

9.99876 

40 

5x6 
515 
515 

641 
642 
642 

15660 
15720 
15780 

21 
22 
23 

8.87995 
8.88161 
8.88326 

8.88120 
8.88287 
8.88453 

1.11880 
1.11713 
1.11547 

9.99875 
9.99874 
9.99873 

39 
38 
37 

515 
514 
514 

643 
644 
644 

15840 
15900 
15960 

24 
25 
26 

8.88490 
8.88654 
8.88817 

8.88618 
8.88783 
8.88948 

1.11382 
1.11217 
1.11052 

9.99872 
9.99871 
9.99870 

36 

35 
34 

514 
513 
513 

645 

646 
646 

16020 
16080 
16140 

27 
28 
29 

8.88980 
8.89142 
8.89304 

8.89111 
8.89274 
8.89437 

1.10889 
1.10726 
1.10563 

9.99869 
9.99868 
9.99867 

32 
31 

513 

647 

16200 

30 

8.89464 

8.89598 

1.10402 

9.99866 

30 

512 
512 
512 

648 
648 
649 

16260 
16320 
16380 

31 
32 
Z2> 

8.89625 
8.89784 
8.89943 

8.89760 
8.89920 
8.90080 

1.10240 
1.10080 
1.09920 

9.99865 
9.99864 
9.99863 

29 

28 
27 

5" 
5" 

650 

650 

651 

16440 
16500 
16560 

34 
35 
36 

8.90102 
8.90260 
8.90417 

8.90240 
8.90399 
8.90557 

1.09760 
1.09601 
1.09443 

9.99862 
9.99861 
9.99860 

26 
25 
24 

510 
510 
510 

652 
652 
653 

16620 
16680 
16740 

37 
38 
39 

8.90574 
8.90730 
8.90885 

8.90715 
8.90872 
8.91029 

1.09285 
1.09128 
1.08971 

9.99859 
9.99858 
9.99857 

23 
22 
21 

509 

654 

16800 

40 

8.91040 

8.91185 

1.08815 

9.99856 

20 

509 
509 
508 

654 
655 
656 

16860 
16920 
16980 

41 
42 
43 

8.91195 
8.91349 
8.91502 

8.91340 
8.91495 
8.91650 

1.08660 
1.08505 
1.08350 

9.99855 
9.99854 
9.99853 

19 
18 
17 

508 
508 
507 

656 
657 
658 

17040 
17100 
17160 

44 
45 
46 

8.91655 
8.91807 
8.91959 

8.91803 
8.91957 
8.92110 

1.08197 
1.08043 
1.07890 

9.99852 
9.99851 
9.99850 

16 
15 
14 

507 

507 
506 

659 

659 

660 

17220 
17280 
17340 

47 
48 
49 

8.92110 
8.92261 
8.92411 

8.92262 
8.92414 
8.92565 

1.07738 
1.07586 
1.07435 

9.99848 
9.99847 
9.99846 

13 
12 
11 

506 

661 

17400 

50 

8.92561 

8.92716 

1.07284 

9.99845 

10 

S06 
505 
505 

661 
662 
663 

17460 
17520 
17580 

51 

52 
53 

8.92710 
8.92859 
8.93007 

8.92866 
8.93016 
8.93165 

1.07134 
1.06984 
1.06835 

9.99844 
9.99843 
9.99842 

9 

8 

7 

505 
504 
504 

664 
664 
665 

17640 
17700 
17760 

54 
55 
56 

8.93154 
8.93301 
8.93448 

8.93313 
8.93462 
8.93609 

1.06687 
1.06538 
1.06391 

9.99841 
9.99840 
9.99839 

6 

5 
4 

503 
503 
503 

666 
666 
667 

17820 
17880 
17940 

57 
58 
59 

8.93594 
8.93740 
8.93885 

8.93756 
8.93903 
8.94049 

1.06244 
1.06097 
1.05951 

9.99838 
9.99837 
9.99836 

3 
2 
1 
0 

502 

668 

18000 

60 

8.94030 

8.94195 

1.05805 

9.99834 

L.  Cos. 

L.  Cot. 

L.  Tan. 

L.  Sin. 

f 

s.f;° 


129 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


L.  Sin. 


8.94  030 


8.94  174 
8.94  317 
8.94  461 
8.94  603 
8.94  746 

8.94  887 

8.95  029 
8.95  170 
8.95  310 


.95  450 


.95  589 
.95  728 
.95  867 
.96  005 
.96  143 
.96  280 
.96  417 
.96  553 
.96  689 


8.96  825 


8.96  960 

8.97  095 
8.97  229 
8.97  363 
8.97  496 
8.97  629 
8.97  762 

8.97  894 

8.98  026 


8.98  157 


8.98  288 
8.98  419 
8.98  549 
8.98  679 
8.98  808 

8.98  937 

8.99  066 
8.99  194 
8.99  322 


8.99  450 


8.99  577 
8.99  704 
8.99  830 
8.99  956 
9.00  082 
9.00  207 
9.00  332 
9.00  456 
9.00  581 


9.00  704 


9.00  828 

9.00  951 

9.01  074 
9.01  196 
9.01  318 
9.01  440 
9.01  561 
9.01  682 
9.01  803 


9.01  923 
L.  Cos. 


L.  Tan.  c.d.  L.  Cot 


8.94  195 


8.94 
8.94 
8.94 
8.94 
8.94 
8.95 
8.95 
8.95 
8.95 


340 
485 
630 
773 
917 
060 
202 
344 
486 


.95  627 


8.95 
8.95 
8.96 
8.96 
8.96 
8.96 
8.96 
8.96 
8.96 


767 
908 
047 
187 
325 
464 
602 
739 
877 


8.97  013 


8.97 
8.97 
8.97 
8.97 
8.97 
8.97 
8.97 
8.98 
8.98 


150 
285 
421 
556 
691 
825 
959 
092 
225 


8.98  358 


8.98 
8.98 
8.98 
8.98 
8.99 
8.99 
8.99 
8.99 
8.99 


490 
622 
753 
884 
015 
145 
275 
405 
534 


8.99  662 


8.99 
8.99 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 


791 
919 
046 
174 
301 
427 
553 
679 
805 


9.00  930 


9.01 
9.01 
9.01 
9.01 
9.01 
9.01 
9.01 
9.01 
9.02 


055 
179 
303 
427 
550 
673 
796 
918 
040 


9.02  162 
L.  Cot. 


145 
145 
145 
143 
144 
143 
142 
142 
142 
141 
140 
141 
139 
140 
138 
139 
138 
137 
138 
136 
137 
135 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
130 
130 
130 
129 
128 
129 
128 
127 
128 
127 
126 
126 
126 
126 
125 
125 
124 
124 
124 
123 
123 
123 
122 
122 
122 


1.05  805 


.05  660 
.05  515 
.05  370 
.05  227 
.05  083 
.04  940 
.04  798 
.04  656 
,04  514 


,04  373 


.04  233 
,04  092 
,03  953 
,03  813 
,03  675 
,03  536 
,03  398 
,03  261 
,03  123 


,02  987 


,02  850 
,02  715 
,02  579 
,02  444 
,02  309 
,02  175 
,02  041 
,01  908 
,01  775 


.01  642 


.01  510 
.01  378 
.01  247 
.01  116 
.00  985 
.00  855 
.00  725 
.00  595 
.00  466 


.00  338 


.00  209 
.00  081 
0.99  954 
0.99  826 
0.99  699 
0.99  573 
0.99  447 
0.99  321 
0.99  195 


0.99  070 


0.98  945 
0.98  821 
0.98  697 
0.98  573 
0.98  450 
0.98  327 
0.98  204 
0.98  082 
0.97  960 


0.97  838 
L.  Tan. 


L.  Cos. 


9.99  834 


9.99  833 
9.99.832 
9.99  831 
9.99  830 
9.99  829 
9.99  828 
9.99  827 
9.99  825 
9.99  824 


9.99  823 


9.99  822 
9.99  821 
9.99  820 
9.99  819 
9.99  817 
9.99  816 
9.99  815 
9.99  814 
9.99  813 


9.99  812 


9.99  810 
9.99  809 
9.99  808 
9.99  807 
9.99  806 
9.99  804 
9.99  803 
9.99  802 
9.99  801 


9.99  800 


9.99  798 
9.99  797 
9.99  796 
9.99  795 
9.99  793 
9.99  792 
9.99  791 
9.99  790 
9.99  788 


9.99  787 


9.99  786 
9.99  785 
9.99  783 
9.99  782 
9.99  781 
9.99  780 
9.99  778 
9.99  777 
9.99  776 


9.99  775 


9.99  773 
9.99  772 
9.99  771 
9.99  769 
9.99  768 
9.99  767 
9.99  765 
9.99  764 
9.99  763 


9.99  761 
L.  Sin.    I 


20 

19 
18 
17 
16 
15 
14 
13 
12 
1]_ 

9 
8 
7 
6 
5 
4 
3 
2 
_1^ 
0 


P.P. 


145 

144 

143 

12. 1 

12.0 

1 1.9 

24.2 
36.2 
48.3 
60.4 

24.0 
36.0 
48.0 
60.0 

23.8 
35.8 
47.7 
59.6 

72.S 

84.6 

96.7 

108.8 

72.0 

84.0 

96.0 

108.0 

71.S 

83.4 

95.3 

107.2 

120.8 

120.0 

119. 2 

132.9 

132.0 

131. 1 

141 


11.8 

23.5 

35.2 

47.0 

58.8 

70.5 

82.2 

94.0 

105.8 

117.S 

129.2 


140 

11.7 

23-3 

35.0 

46.7 

58.3 

70.0 

81.7 

93.3 

1 05.0 

1 16.7 

128.3 


139 

11.6 

23.2 

34.8 

46.3 

57-9 

69.5 

81.1 

92.7 

104.2 

11S.8 

127.4 


137 

136 

135 

1 1.4 

1 1.3 

11.2 

22.8 

22.7 

22.5 

34.2 

34.0 

33.8 

45.7 

45.3 

45.0 

57.1 

.56.7 

.■>6.2 

68.5 

68.0 

67.5 

79.9 

79.3 

78.8 

91.3 

90.7 

90.0 

102.8 

102.0 

101.2 

114.2 

1 13.3 

112.5 

125.6 

124.7 

123.8 

II.8 
23.7 

35'S 

47.3 

59.2 

71.0 

82.8 

94-7 

106.5 

118.3 

130.2 


138 

ii.S 

23.0 

34-5 

46.0 

57.S 

69.0 

80.S 

92.0 

103.5 

iiS.o 

126.S 


11.2 

22.3 

33.5 

44-7 

SS.8 

67.0 

78.2 

89.3 

100.5 

111.7 

122.8 


"I  133     132     131     130 


II. I 

22.2 
33.2 
44.3 
55.4 
66.5 
77-6 
88.7 
99.8 
110.8 
121.9 


II.O 
22.0 
33.0 

44.0 

55.0 
66.0 
77.0 
88.0 
99.0 

IIO.O 


10.9 
21.8 
32.8 
43.7 
54.6 
65.5 
76.4 
87.3 
98.2 
109.2 


10.8 
21.7 
32.5 
43.3 
54.2 
65.0 
75.8 
86.7 
97.5 
108.3 
119.2 


129 

128 

127 

126 

10.8 

10.7 

10.6 

lo.s 

21.5 

21.3 

21.2 

21.0 

32.2 

32.0 

31.8 

31.5 

43.0 

42.7 

42.3 

42.0 

53.8 

53.3 

52.9 

52.5 

64.5 

64.0 

63.5 

63.0 

75.2 

74-7 

74.1 

73.S 

86.0 

85.3 

84.7 

84.0 

96.8 

96.0 

95.2 

94.5 

107.S 

106.7 

105.8 

105.0 

1X8.2 

II7.3 

1 16.4 

115.S 

" 

125 

124 

5 

10.4 

10.3 

10 

20.8 

20.7 

15 

31.2 

31.0 

20 

41.7 

41.3 

25 

52.1 

51.7 

30 

62.5 

62.0 

35 

72.9 

72.3 

40 

83.3 

82.7 

45 

93.8 

93-0 

50 

104.2 

103-3 

55 

1 14.6 

1 13.7 

10.2 

10.2 

20.5 

20.3 

30.8 

30.5 

41.0 

40.7 

51.2 

50.8 

61.5 

61.0 

71.8 

71.2 

82.0 

81.3 

92.2 

9I.S 

102. 5  101.7 
II2.8  III.8 


P.P. 


130 


6° 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot, 


L.  Cos. 


P.P. 


10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

3& 

39_ 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 


50 


51 

52 
53 
54 
55 
56 
57 
58 
59 


60 


9.01  923 


9.02  043 
9.02  163 
9.02  283 
9.02  402 
9.02  520 
9.02  639 
9.02  757 
9.02  874 
9.02  992 


9.03  109 


9.03  226 
9.03  342 
9.03  458 
9.03  574 
9.03  690 
9.03  805 

9.03  920 

9.04  034 
9.04  149 


9.04  262 


9.04  376 
9.04  490 
9.04  603 
9.04  715 
9.04  828 

9.04  940 

9.05  052 
9.05  164 
9.05  275 


9.05  386 


9.05  497 
9.05  607 
9.05  717 
9.05  827 

9.05  937 

9.06  046 
9.06  155 
9.06  264 
9.06  372 


9.06  481 


9.06  589 
9.06  696 
9.06  804 

9.06  911 

9.07  018 
9.07  124 
9.07  231 
9.07  337 
9.07  442 


9.07  548 


9.07  653 
9.07  758 
9.07  863 

9.07  968 

9.08  072 
9.08  176 
9.08  280 
9.08  383 
9.08  486 


9.08  589 


120 
120 
120 
119 
118 
119 
118 
117 
118 
117 
117 
116 
116 
116 
116 
115 
115 
114 
115 
113 
114 
114 
113 
112 
113 
112 
112 
112 
111 
111 
111 
110 
110 
110 
110 
109 
109 
109 
108 
109 
108 
107 
108 
107 
107 
106 
107 
106 
105 
106 
105 
105 
105 
105 
104 
104 
104 
103 
103 
103 


.02  162 


.02  283 
02  404 
.02  525 
.02  645 
02  766 

02  885 

03  005 
03  124 
03  242 


03  361 


03  479 
03  597 
03  714 
03  832 

03  948 

04  065 
04  181 
04  297 
04  413 


04  528 


04  643 
04  758 
04  873 

04  987 

05  101 
05  214 
05  328 
05  441 
05  553 


05  666 


05  778 

05  890 

06  002 
06  113 
06  224 
06  335 
06  445 
06  556 
06  666 


06  775 


06  885 

06  994 

07  103 
07  211 
07  320 
07  428 
07  536 
07  643 
07  751 


07  858 


07  964 

08  071 
08  177 
08  283 
08  389 
08  495 
08  600 
08  705 
08  810 


08  914 


121 
121 
121 
120 
121 
119 
120 
119 
118 
119 
118 
118 
117 
118 
116 
117 
116 
116 
116 
115 
115 
115 
115 
114 
114 
113 
114 
113 
112 
113 
112 
112 
112 
111 
111 
111 
110 
111 
110 
109 
110 
109 
109 
108 
109 
108 
108 
107 
108 
107 
106 
107 
106 
106 
106 
106 
105 
105 
105 
104 


0.97  838 


9.99  761 


0.97  717 
0.97  596 
0.97  475 
0.97  355 
0.97  234 
0.97  115 
0.96  995 
0.96  876 
0.96  758 


0.96  639 


0.96  521 
0.96  403 
0.96  286 
0.96  168 
0.96  052 
0.95  935 
0.95  819 
0.95  703 
0.95  587 


0.95  472 


0.95  357 
0.95  242 
0.95  127 
0.95  013 
0.94  899 
0.94  786 
0.94  672 
0.94  559 
0.94  447 


0.94  334 


0.94  222 
0.94  110 
0.93  998 
0.93  887 
0.93  776 
0.93  665 
0.93  555 
0.93  444 
0.93  334 


0.93  225 


0.93  115 
0.93  006 
0.92  897 
0.92  789 
0.92  680 
0.92  572 
0.92  464 
0.92  357 
0.92  249 


0.92  142 


0.92  036 
0.91  929 
0.91  823 
0.91  717 
0.91611 
0.91  505 
0.91  400 
0.91  295 
0.91  190 


0.91  086 


9.99  760 
9.99  759 
9.99  757 
9.99  756 
9.99  755 
9.99  753 
9.99  752 
9.99  751 
9.99  749 


9.99  748 


9.99  747 
9.99  745 
9.99  744 
9.99  742 
9.99  741 
9.99  740 
9.99  738 
9.99  737 
9.99  736 


9.99  734 


9.99  733 
9.99  731 
9.99  730 
9.99  728 
9.99  727 
9.99  726 
9.99  724 
9.99  723 
9.99  721 


9.99  720 


9.99  718 
9.99  717 
9.99  716 
9.99  714 
9.99  713 
9.99  711 
9.99  710 
9.99  708 
9.99  707 


9.99  705 


9.99  704 
9.99  702 
9.99  701 
9.99  699 
9.99  698 
9.99  696 
9.99  695 
9.99  693 
9.99  692 


9.99  690 


9.99  689 
9.99  687 
9.99  686 
9.99  684 
9.99  683 
9.99  681 
9.99  680 
9.99  678 
9.99  677 


50 


20 


9.99  675 


119 

9.9 
19.8 
29.8 
39.7 
49.6 
59.5 
69.4 
79.3 
89.2 
99.2 
1 09. 1 


118      117     116 


121 

120 

10. 1 

lO.O 

20.2 

20.0 

30.2 

30.0 

40.3 

40.0 

50.4 

So.o 

00.5 

60.0 

70.6 

70.0 

80.7 

80.0 

90.8 

90.0 

100.8 

lOO.O 

1 10.9 

IIO.O 

9.8 

19.7 

29-5 
39-3 
49.2 
59-0 
68.8 
78.7 
88.5 
98.3 
108.2 


9.8 
19-5 
29.2 
39-0 
48.8 
S8.5 
68.2 
78.0 
87.8 
97.5 
107.2 


115 

114 

9.6 

9.5 

19.2 

19.0 

28.8 

28.S 

38.3 

38.0 

47.9 

47.5 

57.5 

S7.0 

67.1 

66.5 

76.7 

76.0 

86.2 

85.5 

95.8 

95-0 

10S.4 

104.5 

112 

111 

9.3 

9.2 

18.7 

la..-; 

28.0 

27.8 

37.3 

37-0 

46.7 

46.2 

56.0 

55.5 

65.3 

64.8 

74-7 

74-0 

84.0 

83.2 

93.3 

92.S 

102.7 

101.8 

9.7 

19-3 
29.0 
38.7 
48.3 
58.0 
67.7 
77.3 
87.0 
96.7 
106.3 


9.4 
18.8 
28.2 
37.7 
47.1 
56.5 
65-9 
75-3 
84.8 
94-2 
103.6 


110 

9.2 
18.3 
27.5 
36.7 
45.8 
S5.0 
64.2 
73-3 
82.S 
91.7 
100.8 


1091  108 


5    9-1     9 

10  18.2J18 
15  27.2,27 
20I36.3  36 
25|4S.4'45 
30  54-5  54 
35,63.663 
40  72.7I72 
45,8i.8|8i 
5090.8  90 
55  99.9'99 


106 

105 

8.8 

8.8 

17.7 

17-5 

26.5 

26.2 

35.3 

35-0 

44.2 

43.8 

53.0 

52.5 

61.8 

61.2 

70.7 

70.0 

79.5 

78.8 

88.3 

87.5 

97.2 

96.2 

107 

o  8.9 
0  17.8 
o  26.8 
035.7 
044.6 
053.5 
062.4 
oi7i.3 

0)80.2 

o  89.2 
o'98.i 


104 

8.7 
17.3 
26.0 
34-7 
43.3 
52.0 
60.7 
09-3 
78.0 
86.7 
95.3 


L.  Cos.      d.      L.  Cot.      c.d.      L.  Tan 


L.  Sin. 


P.P. 


7° 


131 


L.  Sin.   d.   L.  Tan.  c.d.   L.  Cot. 


L.  Cos. 


P.P. 


10 


20. 

21 
22 
23 
24 
25 
26 
27 
28 
29 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


51 

52 
53 
54 
55 
56 
57 
58 

60 


08  589 


08  692 
08  795 
08  897 

08  999 

09  101 
09  202 
09  304 
09  405 
09  506 


9.09  606 


09  707 
09  807 

09  907 

10  006 
10  106 
10  205 
10  304 
10  402 
10  501 


9.10  599 

9 

9 

9 


10  697 
10  795 
10  893 

10  990 
11087 

11  184 
11281 
11377 
11474 


11  666 

11  761 
11857 
11952 

12  047 
12  142 
12  236 
12  331 
12  425 


12  519 


12  612 
12  706 
12  799 
12  892 

12  985 

13  078 
13  171 
13  263 
13  355 


9.13  447 


13  539 
13  630 
13  722 
13  813 
13  904 

13  994 

14  085 
14  175 
14  266 


9.11  570 


14  356 


103 

103 

102 

102 

102 

101 

102 

101 

101 

100 

101 

100 

100 

99 

100 

99 

99 

98 

99 

98 

98 

98 

98 

97 

97 

97 

97 

96 

97 

96 

96 

95 

96 

95 

95 

95 

94 

95 

94 

94 

93 

94 

93 

93 

93 

93 

93 

92 

92 

92 

92 

91 

92 

91 

91 

90 

91 

90 

91 

90 


9.08  914 


9.09  019 
9.09  123 
9.09  227 
9.09  330 
9.09  434 
9.09  537 
9.09  640 
9.09  742 
9.09  845 


9.09  947 


9.10  049 
9.10  150 
9.10  252 
9.10  353 
9.10  454 
9.10  555 
9.10  656 
9.10  756 
9.10  856 


9.10  956 


9.11  056 
9.11  155 
9.11  254 
9.11353 
9.11452 
9.11551 
9.11649 
9.11  747 
9.11845 


9.11943 


12  040 
12  138 
12  235 
12  332 
12  428 
12  525 
12  621 
12  717 


9.12  813 


9.12  909 


9.13  004 

9.13  099 

9.13  194 

9.13  289 

13  384 

13  478 

13  573 

13  667 

13  761 


9.13  854 


9.13  948 

9.14  041 
9.14  134 
9.14  227 
9.14  320 
9.14  412 
9.14  504 
9.14  597 
9.14  688 


9.14  780 


105 
104 
104 
103 
104 
103 
103 
102 
103 
,102 
102 
101 
102 
101 
101 
101 
101 
100 
100 
100 
100 
99 
99 
99 
99 
99 
98 
98 
98 
98 
97 
98 
97 
97 
96 
97 
96 
96 
96 
96 
95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 
93 
91 
92 


0.91  086 


9.99  675 


60 


0.90  981 
0.90  877 
0.90  773 
0.90  670 
0.90  566 
0.90  463 
0.90  360 
0.90  258 
0.90  155 


9.99  674 
9.99  672 
9.99  670 


9.99  669 
9.99  667 
9.99  666 

56 

55 
54 

9.99  664 
9.99  663 
9.99  661 

53 
52 
51 

0.90  053 
0.89  951 
0.89  850 
0.89  748 
0.89  647 
0.89  546 
0.89  445 
0.89  344 
0.89  244 
0.89  144 


9.99  659 
9.99  658 
9.99  656 
9.99  655 
9.99  653 
9.99  651 
9.99  650 
9.99  648 
9.99  647 
9.99  645 


50 


0.89  044 


9.99  643 


0.88  944 
0.88  845 
0.88  746 
0.88  647 
0.88  548 
0.88  449 
0.88  351 
0.88  253 
0.88  155 


9.99  642 
9.99  640 
9.99  638 
9.99  637 
9.99  635 
9.99  633 
9.99  632 
9.99  630 
9.99  629 


0.88  057 


9.99  627 


0.87  960 
0.87  862 
0.87  765 
0.87  668 
0.87  572 
0.87  475 
0.87  379 
0.87  283 
0.87  187 


9.99  625 
9.99  624 
9.99  622 
9.99  620 
9.99  618 
9.99  617 
9.99  615 
9.99  613 
9.99  612 


0.87  091 


9.99  610 


20 


0.86  996 
0.86  901 
0.86  806 
0.86  711 
0.86  616 
0.86  522 
0.86  427 
0.86  333 
0.86  239 


9.99  608 
9.99  607 
9.99  605 
9.99  603 
9.99  601 
9.99  600 
9.99  598 
9.99  596 
9.99  595 


0.86  146 


9.99  593 


0.86  052 
0.85  959 
0.85  866 
0.85  773 
0.85  680 
0.85  588 
0.85  496 
0.85  403 
0.85  312 


9.99  591 
9.99  589 
9.99  588 
9.99  586 
9.99  584 
9.99  582 
9.99  581 
9.99  579 
9.99  577 


0.85  220 


9.99  575 


104!  103   102 


5\  8 
io|i7 
15  26 
2034 
25  43 
3o|S2 
35i6o 
40 1 69 
4578 
5086 
SSI95 


.7  8.6  8.5 
.3;i7-2  17-0 
.0  25.8  25.S 
•7  34-3  34-0 
.342.942.5 
•0  51.S  Si.o 
.7!6o.i  59.5 
.3  68.7  68.0 
.0  77.2  76.S 
.7  85.8  85.0 
•394.493.5 


1011  lOOi    99 


5  8.4  8 
io!i6.8  16 
is'25.2125 
20  33.7[33 
2542.1  41 
30  50.5  50 
3558.958 
40  67.3  66 
45  75.8  75 
SO  84.2  83 
55  92.6  91 


.3!  8.2 
•7  16.5 
.0  24.8 
■333.0 
.7:41.2 
.049.5 
.3  57.8 
.7  66.0 
.0  74.2 
.382.5 
.790.8 


S    8.2 
io|i6.3 

I5;24.5 

20  32.7 
25  40.8 
3049.0 

35  57.2 
40  65.3 
45  73.5 
5081.7 
5589.8 


97  I  96 
8.1 i  8.0 
16.2  16.0 
24.2  24.0 
32.3I32.0 
40. 4 1 40.0 
48.5148.0 
56.6  56.0 
64.7164.0 
72.8  72.0 
180.880.0 
l88.9'88.o 


95  I  94  93 


7.8 
15-5 
23.2 


20  31 
25  39 
3047 
3S!55 
4063 


.9  7.8 
•8  15.7 
.823.5 
.7  31.3  31.0 
.6  39.2  38.8 
.5  47-0  46.5 
.4  54.8  54-2 


62.7  62.0 

70.5  69.8 

78.3  77.5 

I  86.2,85.2 


92  I  91  I  90 


5  7.7 
10,15.3 
15  23.0 
20130.7 
25  38.3 
30  46.0 
35  53-7 
40  61.3 
45  69.0 
50  76.7 
55  84.3 


7.6  75 
15.2:15.0 
22.8  22.5 
30.3:30.0 
37-9  37.5 
45-5  45.0 
53-1  52.5 

60.7  60.0 
68.2  67.5 

75.8  75.0 
83.4  82.5 


L.  Cos.       d. 


L.  Cot. 


c.d. 


L.  Tan. 


82° 


L.  Sin. 


P.P. 


132 

8° 

' 

L.  Sin. 

d. 

L.  Tan.  |  c.d. 

L.  Cot. 

L.  Cos. 

P.P. 

0 

9.14  356 

89 
90 
89 
90 
89 
88 
89 
89 
88 
88 
88 
88 
87 
88 
87 
87 
87 
87 
86 
86 
87 
86 
85 

9.14  780 

92 
91 
91 
91 
91 
91 
90 
91 
90 
90 
89 
90 
89 
90 
89 
89 
88 
89 
88 
88 
88 
88 
88 

0.85  220 

9.99  575 

60 

1 

9.14  445 

9.14  872 

0.85  128 

9.99  574 

59 

2 

9.14  535 

9.14  963 

0.85  037 

9.99  572 

58 

3 

9.14  624 

9.15  054 

0.84  946 

9.99  570 

57 

"  j  91  90  89 

4 

9.14  714 

9.15  145 

0.84  855 

9.99  568 

56 

5  7.6  7-5  7-4 

5 

9.14  803 

9.15  236 

0.84  764 

9.99  566 

55 

lojis.a  I5.0|i4.8 

ISj22.8|22.Sj22.2 

20|30.3[30.0  29.7 

6 

9.14  891 

9.15  327 

0.84  673 

9.99  565 

54 

7 

9.14  980 

9.15  417 

0.84  583 

9.99  563 

53 

25!37.9'37.Sl37.l 
■30  45-5  45.0  44-5 
35  53-1  52. 5  51-9 

8 

9.15  069 

9.15  508 

0.84  492 

9.99  561 

52 

9 

9.15  157 

9.15  598 
9.15  688 
9.15  777 

0.84  402 

9.99  559 

51 

40  60.7  60.0  59-3 
45  68.2  67. 5  66.8 
50:75.8,75.0  74-2 
SS83.482.S  81.6 

10 

11 

9.15  245 

0.84  312 

9.99  557 

50 

9.15  333 

0.84  223 

9.99  556 

49 

12 

9.15  421 

9.15  867 

0.84  133 

9.99  554 

48 

13 

9.15  508 

9.15  956 

0.84  044 

9.99  552 

47 

14 

9.15  596 

9.16  046 

0.83  954 

9.99  550 

46 

15 

9.15  683 

9.16  135 

0.83  865 

9.99  548 

45 

16 

9.15  770 

9.16  224 

0.83  776 

9."^9  546 

44 

17 

9.15  857 

9.16  312 

0.83  688 

9.99  545 

43 

18 

9.15  944 

9.16  401 

0.83  599 

9.99  543 

42 

19 

9.16  030 

9.16  489 

0.83  511 

9.99  541 

41 

5 
10 

»»   87   Ht> 

7.3  7.2  7.2 
14.7  14-5  14.3 

20 

9.16  116 

9.16  577 

0.83  423 
0.83  335 

9.99  539 

40 

21 

9.16  203 

9.16  665 

9.99  537 

39 

I5|22.0;2I.8|2I.S 

2029.329.028.7 
2S'36.7  36.235.8 

22 

9.16  289 

9.16  753 

0.83  247 

9.99  535 

38 

23 

9.16  374 

9.16  841 

0.83  159 

9.99  533 

37 

30I44.0143.5  43.0 

24 

9.16  460 

86 
85 
86 
85 
85 
85 
84 
85 
84 
84 
84 
84 
83 

9.16  928 

8'/ 
88 
87 
87 
87 
86 
87 
86 
86 
86 
86 
86 
85 

0.83  072 

9.99  532 

36 

35  51.3  50.8  50.2 
40  58.7  58.0  57.3 

75 

9.16  545 

9.17  016 

0.82  984 

9.99  530 

35 

45  66.0  65.2  64.S 

26 

9.16  631 

9.17  103 

0.82  897 

9.99  528 

34 

50  73.3  72.5  71-7 
55  80.7I79.8  78.8 

27 

9.16  716 

9.17  190 

0.82  810 

9.99  526 

33 

28 

9.16  801 

9.17  277 

0.82  723 

9.99  524 

31 

29 

9.16  886 

9.17  363 

0.82  637 

9.99  522 
9.99  520 

31 
30 

30 

9.16  970 

9.17  450 

0.82  550 

31 

9.17  055 

9.17  536 

0.82  464 

9.99  518 

29 

32 

9.17  139 

9.17  622 

0.82  378 

9.99  517 

28 

33 

9.17  223 

9.17  708 

0.82  292 

9.99  515 

27 

34 

9.17  307 

9.17  794 

0.82  206 

9.99  513 

26 

// 

85  84  1  83 

35 

9.17  391 

9.17  880 

0.82  120 

9.99  511 

25 

5 

7.1  7.0  6.9 

36 

9.17  474 

9.17  965 

0.82  035 

9.99  509 

24 

10 

14.2  14.0  13.8 

37 

9.17  558 

84 
83 
83 
83 
83 
83 
82 
82 
83 
82 
81 
82 
82 
81 
81 
81 
81 
81 
81 
80 
80 
80 
80 
80 

9.18  051 

86 
85 
85 
85 
85 
84 
85 
84 
84 
84 
84 
83 
84 
83 
83 
83 
S3 
83 
S3 
s? 

0.81  949 

9.99  507 

23 

15  21.2  21.0  20.8 
20  28.3  28.o!27.7 

38 

9.17  641 

9.18  136 

0.81  864 

9.99  505 

22 

2S'35.4|35.0  34-6 

39 
40 

9.17  724 

9.18  221 

0.81  779 
0.81  694 

9.99  503 

21 

30 
35 
40 
45 
50 

42.5 
49.6 
56.7 
63.8 
70.8 

42.041.5 
49.o'48.4 
56.055.3 
63.0  62.2 
70.0  69.2 

9.17  807 

9.18  306 

9.99  501 

20 

41 

9.17  890 

9.18  391 

0.81  609 

9.99  499 

19 

42 

9.17  973 

9.18  475 

0.81  525 

9.99  497 

18 

S5l77.9'77.0  76.1 

43 

9.18  055 

9.18  560 

0.81  440 

9.99  495 

17 

44 

9.18  137 

9.18  644 

0.81  356 

9.99  494 

16 

45 

9.18  220 

9.18  728 

0.81  272 

9.99  492 

15 

46 

9.18  302 

9.18  812 

0.81  188 

9.99  490 

14 

47 

9.18  383 

9.18  896 

0.81  104 

9.99  488 

13 

48 

9.18  465 

9.18  979 

0.81  021 

9.99  486 

12 

49 
50 

9.18  547 

9.19  063 

0.80  937 

9.99  484 

11 

5 

82 

6.8 

81,80 

6.8  6.7 
I3.5ii3.3 
20.2  20.0 

9.18  628 

9.19  146 

0.80  854 

9.99  482 

10 

51 

9.18  709 

9.19  229 

0.80  771 

9.99  480 

9 

15  20.5 

52 

9.18  790 

9.19  312 

0.80  688 

9.99  478 

8 

20  27.3  27.026.7 

53 

9.18  871 

9.19  395 

0.80  605 

9.99  476 

7 

25,34-2  33.8  33.3 
30  41.0  40.5  40.0 

54 

9.18  952 

9.19  478 

0.80  522 

9.99  474 

6 

35  47.8  47.2'46.7 

55 

9.19  033 

9.19  561 

0.80  439 

9.99  472 

5 

40  54-7  54o'S3.3 
45  61. 5  60.8  60.0 

56 

9.19  113 

9.19  643 

82 
82 

82 

0.80  357 

9.99  470 

4 

50  68.3  67.S  66.7 

57 

9.19  193 

9.19  725 

0.80  275 

9.99  468 

3 

55  75.2  74-2  73.3 

58 

9.19  273 

9.19  807 

0.80  193 

9.99  466 

2 

59 
60 

9.19  353 

9.19  889 

82 

0.80  111 

9.99  464 

1 

9.19  433 

9.19  971 

0.80  029 

9.99  462 

0 

L.  Cos.  1  d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

/ 

P.P.           i 

81 


9° 


133 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot. 


L.  Cos. 


P.P. 


10 


11 
12 
13 
14 
15 
16 
17 
18 
19 

21 
22 
23 
24 
25 
26 
27 
28 

_30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.19  433 


9.19  513 
9.19  592 
9.19  672 
9.19  751 
9.19  830 
9.19  909 

9.19  988 

9.20  067 
9.20  145 


9.20  223 


9.20  302 
9.20  380 
9.20  458 
9.20  535 
9.20  613 
9.20  691 
9.20  768 
9.20  845 
9.20  922 


9.20  999 


9.21076 
9.21  153 
9.21  229 
9.21  306 
9.21  382 
9.21  458 
9.21  534 
9.21  610 
9.21  685 


9.21  761 


9.21  836 
9.21912 

9.21  987 

9.22  062 
9.22  137 
9.22  211 
9.22  286 
9.22  361 
9.22  435 


9.22  509 


9.22  583 
9.22  657 
9.22  731 
9.22  805 
9.22  878 

9.22  952 

9.23  025 
9.23  098 
9.23  171 


9.23  244 


9.23  317 
9.23  390 
9.23  462 
9.23  535 
9.23  607 
9.23  679 
9.23  752 
9.23  823 
9.23  895 


9.23  967 


80 
79 

80 
79 
79 
79 
79 
79 
78 
78 
79 
78 
78 
77 
78 
78 
77 
77 
77 
77 
77 
77 
76 
77 
76 
76 
76 
76 
75 
76 
75 
76 
75 
75 
75 
74 
75 
75 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 
73 
73 
73 
73 
72 
73 
72 
72 
73 
71 
72 
72 


9.19  971 


9.20  053 
9.20  134 
9.20  216 
9.20  297 
9.20  378 
9.20  459 
9.20  540 
9.20  621 
9.20  701 


9.20  782 


9.20  862 

9.20  942 

9.21  022 
9.21  102 
9.21  182 
9.21  261 
9.21  341 
9.21  420 
9.21  499 


9.21  578 


9.21  657 
9.21  736 
9.21  814 

9.21  893 
9.21971 

9.22  049 
9.22  127 
9.22  205 
9.22  283 


9.22  361 


9.22  438 
9.22  516 
9.22  593 
9.22  670 
9.22  747 
9.22  824 
9.22  901 

9.22  977 

9.23  054 


9.23  130 


9.23  206 
9.23  283 
9.23  359 
9.23  435 
9.23  510 
9.23  586 
9.23  661 
9.23  737 
9.23  812 


9.23  887 


9.23  962 

9.24  037 
9.24  112 
9.24  186 
9.24  261 
9.24  335 
9.24  410 
9.24  484 
9.24  558 


9.24  632 


82 
81 
82 
81 
81 
81 
81 
81 
80 
81 
80 
80 
80 
80 
80 
79 
80 
79 
79 
79 
79 
79 
78 
79 
78 
78 
78 
78 
78 
78 
77 
78 
77 
77 
77 
77 
77 
76 
77 
76 
76 
77 
76 
76 
75 
76 
75 
76 
75 
75 
75 
75 
75 
74 
75 
74 
75 
74 
74 
74 


0.80  029 


9.99  462 


60 


0.79  947 
0.79  866 
0.79  784 
0.79  703 
0.79  622 
0.79  541 
0.79  460 
0.79  379 
0.79  299 


9.99  460 
9.99  458 
9.99  456 
9.99  454 
9.99  452 
9.99  450 
9.99  448 
9.99  446 
9.99  444 


0.79  218 


9.99  442 


0.79  138 
0.79  058 
0.78  978 
0.78  898 
0.78  818 
0.78  739 
0.78  659 
0.78  580 
0.78  501 


0.78  422 


9.99  440 
9.99  438 
9.99  436 
9.99  434 
9.99  432 
9.99  429 
9.99  427 
9.99  425 
9.99  423 
9.99  421 


40 


0.78  343 
0.78  264 
0.78  186 
0.78  107 
0.78  029 
0.77  951 
0.77  873 
0.77  795 
0.77  717 


9.99  419 
9.99  417 
9.99  415 
9.99  413 
9.99  411 
9.99  409 
9.99  407 
9.99  404 
9.99  402 


0.77  639 


0.77  562 
0.77  484 
0.77  407 
0.77  330 
0.77  253 
0.77  176 
0.77  099 
0.77  023 
0.76  946 


9.99  398 
9.99  396 
9.99  394 


0.76  870 


9.99  379 


0.76  794 
0.76  717 
0.76  641 
0.76  565 
0.76  490 
0.76  414 
0.76  339 
0.76  263 
0.76  188 


9.99  377 
9.99  375 
9.99  372 
9.99  370 
9.99  368 
9.99  366 
9.99  364 
9.99  362 
9.99  359 


0.76  113 


9.99  357 


0.76  038 
0.75  963 
0.75  888 
0.75  814 
0.75  739 
0.75  665 
0.75  590 
0.75  516 
0.75  442 


9.99  355 
9.99  353 
9.99  351 
9.99  348 
9.99  346 
9.99  344 
9.99  342 
9.99  340 
9.99  337 


0.75  368 


9.99  335 


9.99  400   30 


9.99  392 
9.99  390 
9.99  388 

26 

25 
24 

9.99  385 
9.99  383 
9.99  381 

23 
22 
21 

L.  Cos. 


d.   L.  Cot.  c.d 


L.  Tan. 


L.  Sin. 


39 

38 
37 
36 
35 
34 
33 
31 
31 


"  i  82 
s!  6.8! 
10|I3.7 
15^20.5 
20  27.3 
25  34-2 
30  41-0 
35  47-8 
40  54-7 
4561.5 
50;68.3, 
55I75.2I 


81  I  80 

6.81  6.7 
i3S|i3-3 
20.2  20.0 
27.0J26.7 
33.8j33.3 
40.5  40.0 
47.2  46.7 
54-0  53-3 
60.8  60.0 
67-5  66.7 
74.2173.3 


79     78  I  77 


5!  6.6 
1013.2 
15  19.8 

20;26.3 
2S'32.9 

30J39.5 
35146.1 
40,52.7 

45  59-2 
5065.8 
5572.4 


6.5  6.4 
13.0  12.8 
I9.5'i9.2 
26.0  25.7 
32.S  32.1 
39.0  38.5 
45-5  44-9 
52.0  51-3 
58.5  57.8 
65.0164.2 
7 1. 5 1 70.6 


'     76  I  75  I  74 


3  6.2  6.2 
7112.5  12.3 
o  18.8I18.S 
25.0  24.7 
31.2  30.8 
37.537.0 
43.843.2 
50.0  49.3 


56.255.5 
62.5161.7 
68.8167.8 


S  6.1' 
10  12.2 
15I18.2 
20  24.3 
25  30.4 
3036. 5 
35-42.6 
40148.7 
4554.8 
50  60.8 
55166.9 


72     71 

6.0  5.9 
12.0  11.8 
18.0  17.8 
24.0  23.7 
30.o|29.6 
36.0)35.5 
42.0  41.4 
48.047.3 
54.0  53-2 
60.0  59.2 
66.oi65.i 


P.P. 


kO° 


134 


10° 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


9.23  967 


9.24  039 
9.24  110 
9.24  181 
9.24  253 
9.24  324 
9.24  395 
9.24  466 
9.24  536 
9.24  607 


9.24  677 


9.24  748 
9.24  818 
9.24  888 

9.24  958 

9.25  028 
9.25  098 
9.25  168 
9.25  237 
9.25  307 


9.25  376 


9.25  445 
9.25  514 
9.25  583 
9.25  652 
9.25  721 
9.25  790 
9.25  858 
9.25  927 
9.25  995 


9.26  063 


9.26  131 
9.26  199 
9.26  267 
9.26  335 
9.26  403 
9.26  470 
9.26  538 
9.26  605 
9.26  672 


9.26  739 


9.26  806 
9.26  873 

9.26  940 

9.27  007 
9.27  073 
9.27  140 
9.27  206 
9.27  273 
9.27  339 


9.27  405 


9.27  471 
9.27  537 
9.27  602 
9.27  668 
9.27  734 
9.27  799 
9.27  864 
9.27  930 
9.27  995 


9.28  060 


L.  Cos.   d. 


72 
71 
71 
72 
71 
71 
71 
70 
71 
70 
71 
70 
70 
70 
70 
70 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 
68 
69 
68 
68 
68 
68 
68 
68 
68 
67 
68 
67 
67 
67 
67 
67 
67 
67 
66 
67 
66 
67 
66 
66 
66 
66 
65 
66 
66 
65 
65 
66 
65 
65 


9.24  632 


9.24  706 
9.24  779 
9.24  853 

9.24  926 

9.25  000 
9.25  073 
9.25  146 
9.25  219 
9.25  292 


9.25  365 


9.25  437 
9.25  510 
9.25  582 
9.25  655 
9.25  727 
9.25  799 
9.25  871 

9.25  943 

9.26  015 


9.26  086 


9.26  158 
9.26  229 
9.26  301 
9.26  372 
9.26  443 
9.26  514 
9.26  585 
9.26  655 
9.26  726 


9.26  797 


9.26  867 

9.26  937 

9.27  008 
9.27  078 
9.27  148 
9.27  218 
9.27  288 
9.27  357 
9.27  427 


9.27  496 


9.27  566 
9.27  635 
9.27  704 
9.27  773 
9.27  842 
9.27  911 

9.27  980 

9.28  049 
9.28  117 


9.28  186 


9.28  254 
9.28  323 
9.28  391 

9.28  459 
9.28  527 
9.28  595 
9.28  662 
9.28  730 
9.28  798 


9.28  865 


74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 
73 
72 
73 
72 
72 
72 
72 
72 
71 
72 
71 
72 
71 
71 
71 
71 
70 
71 
71 
70 
70 
71 
70 
70 
70 
70 
69 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 
68 
69 
68 
69 
68 
68 
68 
68 
67 
68 
68 
67 


0.75  368 


0.75  294 
0.75  221 
0.75  147 
0.75  074 
0.75  000 
0.74  927 
0.74  854 
0.74  781 
0.74  708 


0.74  635 


0.74  563 
0.74  490 
0.74  418 
0.74  345 
0.74  273 
0.74  201 
0.74  129 
0.74  057 
0.73  985 


0.73  914 


0.73  842 
0.73  771 
0.73  699 
0.73  628 
0.73  557 
0.73  486 
0.73  415 
0.73  345 
0.73  274 


0.73  203 


0.73  133 
0.73  063 
0.72  992 
0.72  922 
0.72  852 
0.72  782 
0.72  712 
0.72  643 
0.72  573 


0.72  504 


0.72  434 
0.72  365 
0.72  296 
0.72  227 
0.72  158 
0.72  089 
0.72  020 
0.71  951 
0.71  883 


0.71  814 


0.71  746 
0.71  677 
0.71  609 
0.71  541 
0.71  473 
0.71  405 
0.71  338 
0.71  270 
0.71  202 


0.71  135 


L.  Cot.  c.d.  L.  Tan.  L.  Sin.   d. 


9.99  335 


9.99  333 
9.99  331 
9.99  328 
9.99  326 
9.99  324 
9.99  322 
9.99  319 
9.99  317 
9.99  315 


9.99  313 


9.99  310 
9.99  308 
9.99  306 
9.99  304 
9.99  301 
9.99  299 
9.99  297 
9.99  294 
9.99  292 


9.99  290 


9.99  288 
9.99  285 
9.99  283 
9.99  281 
9.99  278 
9.99  276 
9.99  274 
9.99  271 
9.99  269 


9.99  267 


9.99  264 
9.99  262 
9.99  260 
9.99  257 
9.99  255 
9.99  252 
9.99  250 
9.99  248 
9.99  245 


9.99  243 


9.99  241 
9.99  238 
9.99  236 
9.99  233 
9.99  231 
9.99  229 
9.99  226 
9.99  224 
9.99  221 


9.99  219 


9.99  217 
9.99  214 
9.99  212 

9.99  209 
9.99  207 
9.99  204 
9.99  202 
9.99  200 
9.99  197 


9.99  195 


P.P. 


" 

74 

73  1 

Sj  6.2  6.1 

lO  12.3  12.2 

IS  i8.s  i8.2 

20 

24.7,24.3 

25 

30.8  30.4 

30 

37.036.5 

35  43.2^42.6 

40:49.3 

48.7 

45  55-5 

54-8 

SO  61.7 

60.8 

55 

67.8 

66.9 

72 
6.0 
12.0 
18.0 
24.0 
30.0 
36.0 
42.0 
48.0 
54.0 
60.0 
66.0 


71     70 


5    5 
10  II 

15I17. 
20I23. 
25  29. 
3035. 
35  41. 


5.8 
ii.S 
17.2 
23.0 

28.8 


5-8 

11.7 

17-5 

23.3 

29.2 

35.034.5 

40.8  40.2 
3'46.7  46.0 
2  52.5  51.8 
2  58.3  57-5 
I '64.2  63.2 


30,34 
35  39 
404s 
45  51 
SO  56 
SSI62 


67  I  66 

5.6    5.5 

11.2  II.O 
16.8J16.5 

22.3  22.0 
27.9  27.S 
33.5  33.0 
39-1  38.5 

44.7  44-0 
50.2149.5 

55.8  55-0 

61.4  60.5 


65  I  3  I  2 


5  5-4 
10  10.8 
I5ii6.2 
20  21.7, 
25|27.i 
3o;32.5' 
35  37-9| 
4043.3' 
4548.81 
SO  54.2 
55  59.61 


lAI 


0.2J0.2 
0.5  0.3 

0.8,0.5 
i.o  0.7 


2.5)1.7 
2.8  1.8 


P.P. 


n/Cko 


11 


135 


10 


11 

12 
13 
14 
15 
16 
17 
18 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


9.28  060 


9.28  125 
9.28  190 
9.28  254 
9.28  319 
9.28  384 
9.28  448 
9.28  512 
9.28  577 
9.28  641 


9.28  705 


9.28  769 
9.28  833 
9.28  896 

9.28  960 

9.29  024 
9.29  087 
9.29  150 
9.29  214 
9.29  277 


9.29  340 


9.29  403 
9.29  466 
9.29  529 
9.29  591 
9.29  654 
9.29  716 
9.29  779 
9.29  841 
9.29  903 


9.29  966 


9.30  028 
9.30  090 
9.30  151 
9.30  213 
9.30  275 
9.30  336 
9.30  398 
9.30  459 
9.30  521 


9.30  582 


9.30  643 
9.30  704 
9.30  765 
9.30  826 
9.30  887 

9.30  947 

9.31  008 
9.31  068 
9.31  129 


9.31  189 


9.31  250 
9.31310 
9.31  370 
9.31  430 
9.31  490 
9.31  549 
9.31  609 
9.31  669 
9.31  728 


9.31  788 


L.  Cos. 


65 
65 
64 
65 
65 
64 
64 
65 
64 
64 
64 
64 
63 
64 
64 
63 
63 
64 
63 
63 
63 
63 
63 
62 
63 
62 
63 
62 
62 
63 
62 
62 
61 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
61 
60 
60 
60 
60 
59 
60 
60 
59 
60 


9.28  865 


9.28 
9.29 
9.29 
9.29 
9.29 
9.29 
9.29 
9.29 
9.29 


933 
000 
067 
134 
201 
268 
335 
402 
468 


9.29  535 


9.29 
9.29 
9.29 
9.29 
9.29 
9.29 
9.29 
9.30 
9.30 


601 
668 
734 
800 
866 
932 
998 
064 
130 


9.30  195 


9.30 
9.30 
9.30 
9.30 
9,30 
9.30 
9.30 
9.30 
9.30 


261 
326 
391 

457 
522 
587 
652 
717 
782 


9.30  846 


9.30 
9.30 
9.31 
9.31 
9.31 
9.31 
9.31 
9.31 
9.31 


911 
975 
040 
104 
168 
233 
297 
361 
425 


9.31  489 


9.31 
9.31 
9.31 
9.31 
9.31 
9.31 
9.31 
9.31 
9.32 


552 
616 
679 
743 
806 
870 
933 
996 
059 


9.32  122 


9.32 
9.32 
9.32 
9.32 
9.32 
9.32 
9.32 
9.32 
9.32 


185 
248 
311 
373 
436 
498 
561 
623 
685 


9.32  747 


d.   L.  Cot.   c.d 


68 
67 
67 
67 
67 
67 
67 
67 
66 
67 
66 
67 
66 
66 
66 
66 
66 
66 
66 
65 
66 
65 
65 
66 
65 
65 
65 
65 
65 
64 
65 
64 
65 
64 
64 
65 
64 
64 
64 
64 
63 
64 
63 
64 
63 
64 
63 
63 
63 
63 
63 
63 
63 
62 
63 
62 
63 
62 
62 
62 


0.71  135 


0.71  067 
0.71  000 
0.70  933 
0.70  866 
0.70  799 
0.70  732 
0.70  665 
0.70  598 
0.70  532 


0.70  465 


0.70  399 
0.70  332 
0.70  266 
0.70  200 
0.70  134 
0.70  068 
0.70  002 
0.69  936 
0.69  870 


0.69  805 


0.69  739 
0.69  674 
0.69  609 
0.69  543 
0.69  478 
0.69  413 
0.69  348 
0.69  283 
0.69  218 


0.69  154 


0.69  089 
0.69  025 
0.68  960 
0.68  896 
0.68  832 
0.68  767 
0.68  703 
0.68  639 
0.68  575 


0.68  511 


0.68  448 
0.68  384 
0.68  321 
0.68  257 
0.68  194 
0.68  130 
0.68  067 
0.68  004 
0.67  941 


0.67  878 


0.67  815 
0.67  752 
0.67  689 
0.67  627 
0.67  564 
0.67  502 
0.67  439 
0.67  377 
0.67  315 


0.67  253 


9.99  195 


9.99  192 
9.99  190 
9.99  187 
9.99  185 
9.99  182 
9.99  180 
9.99  177 
9.99  175 
9.99  172 


9.99  170 


9.99  167 
9.99  165 
9.99  162 
9.99  160 
9.99  157 
9.99  155 
9.99  152 
9.99  150 
9.99  147 


9.99  145 


9.99  142 
9.99  140 
9.99  137 
9.99  135 
9.99  132 
9.99  130 
9.99  127 
9.99  124 
9.99  122 


9.99  119 


9.99  117 
9.99  114 
9.99  112 
9.99  109 
9.99  106 
9.99  104 
9.99  101 
9.99  099 
9.99  096 


9.99  093 


9.99  091 
9.99  088 
9.99  086 
9.99  083 
9.99  080 
9.99  078 
9.99  075 
9.99  072 
9.99  070 


9.99  067 


9.99  064 
9.99  062 
9.99  059 
9.99  056 
9.99  054 
9.99  051 
9.99  048 
9.99  046 
9.99  043 


9.99  040 


L.  Tan.   L.  Sin. 


78° 


d. 


^0 

59" 

58 
57 
56 
55 
54 
53 
52 

AL 

50 

49 
48 
47 
46 
45 
44 
43 
42 

il 
JO 

39 

38 
37 
36 
35 
34 
33 
31 

IL 

30 


P.P. 


5  5-7 
10  II.3 
IS  17-0 


67  I 

5.6 


II.2  II.O 

i6.8!i6.s 

20  22.7:22.3'22.0 

25  28.3  27.9  27.5 
30  34-0  33-5  33-0 
35,39-7  39-1  38.5 
40  45.3  44.7J44.0 
45|5i.o  50.2  49.5 
50  56.7  55-8  55.0 
55162.361.4160.5 


65     64  I  63 


5|  54 
10  10.8 
15  16.2 
20  21.7 
25  27.1 
3032.5 
35  37.9' 
40'43.3i 
45|48.8 
5054.2I 
55  59.6' 


5-3  5-2 
10.7,10.5 
i6.o;i5.8 
21.3  21.0 
26.7  26.2 
32.0  31.5 
37.3,36.8 
42.7I42.0 
48.0J47.2 
53.3  52.5 
58.7IS7.8 


62     61     60 


5  5-2 
10  10.3 
I5;i5.5 
20  20.7 
2525.8 
3031.0 
35  36.2 
4041.3 
4546.5 


50 
lO.O 

15.0 


S.I 
10.2 

15-2 

20.3  20.0 

25.4  25.0 

30.5  30.0 
35.635.0 
40.7  40.0 
45.845.0 

50  5 1. 7  50.8  50.0 
5556.8155.955.0 


"I  59 

5  4.9 
10  9.8 
15  14-8 
20  19.7 
24.6 
29.5 
34-4 
39.3 
4S[44.2 
50  49.2 
55-54.1 


3j2 

0.2  0.2 
0.50.3 

0.8  0.5 
i.o  0.7 

1.2  0.8 
1.5  I.O 
l.8;l.2 
2.0  1.3 
2.2  I.S 
2.5  1.7 
2.8  1.8 


P.P. 


136 


12 


1 

2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d 


9.31  788 


9.31  847 
9.31  907 

9.31  966 

9.32  025 
9.32  084 
9.32  143 
9.32  202 
9.32  261 
9.32  319 


9.32  378 


9.32  437 
9.32  495 
9.32  553 
9.32  612 
9.32  670 
9.32  728 
9.32  786 
9.32  844 
9.32  902 


9.32  960 


9.33  018 
9.33  075 
9.33  133 
9.33  190 
9.33  248 
9.33  305 
9.33  362 
9.33  420 
9.33  477 


9.33  534 


9.33  591 
9.33  647 
9.33  704 
9.33  761 
9.33  818 
9.33  874 
9.33  931 

9.33  987 

9.34  043 


9.34  100 


9,34  156 
9.34  212 
9.34  268 
9.34  324 
9.34  380 
9.34  436 
9.34  491 
9.34  547 
9.34  602 


9.34  658 


9.34  713 
9.34  769 
9.34  824 
9.34  879 
9.34  934 

9.34  989 

9.35  044 
9.35  099 
9.35  154 


9.35  209 


59 

60 

59 

59 

59 

59 

59 

59 

58 

59 

59 

58 

58 

59 

58 

58 

58 

58 

58 

58 

58 

57 

58 

57 

58 

57 

57 

58 

57 

57 

57 

56 

57 

57 

57 

56 

57 

56 

56 

57 

56 

56 

56 

56 

56 

56 

55 

56 

55 

56 

55 

56 

55 

55 

55 

55 

55 

55 

55 

55 


9.32  747 


9.32  810 
9.32  872 
9.32  933 

9.32  995 

9.33  057 
9.33  119 
9.33  180 
9.33  242 
9.33  303 


9.33  365 


9.33  426 
9.33  487 
9.33  548 
9.33  609 
9.33  670 
9.33  731 
9.33  792 
9.33  853 
9.33  913 


9.33  974 


9.34  034 
9.34  095 
9.34  155 
9.34  215 
9.34  276 
9.34  336 
9.34  396 
9.34  456 
9.34  516 


9.34  576 


9.34  635 
9.34  695 
9.34  755 
9.34  814 
9.34  874 
9.34  933 

9.34  992 

9.35  051 
9.35  111 


9.35  170 


9.35  229 
9.35  288 
9.35  347 
9.35  405 
9.35  464 
9.35  523 
9.35  581 
9.35  640 
9.35  698 


9.35  757 


9.35  815 
9.35  873 
9.35  931 

9.35  989 

9.36  047 
9.36  105 
9.36  163 
9.36  221 
9.36  279 


9.36  336 


L.  Cos.   d.   L.  Cot.  c.d 


63 
62 
61 
62 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
60 
61 
60 
60 
60 
60 
60 
59 
60 
60 
59 
60 
59 
59 
59 
60 
59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 


0.67  253 


0.67  190 
0.67  128 
0.67  067 
0.67  005 
0.66  943 
0.66  881 
0.66  820 
0.66  758 
0.66  697 


0.66  635 


0.66  574 
0.66  513 
0.66  452 
0.66  391 
0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.66  087 


0.66  026 


0.65  966 
0.65  905 
0.65  845 
0.65  785 
0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 


0.65  424 


0.65  365 
0.65  305 
0.65  245 
0.65  186 
0.65  126 
0.65  067 
0.65  008 
0.64  949 
0.64  889 


0.64  830 


0.64  771 
0.64  712 
0.64  653 
0.64  595 
0.64  536 
0.64  477 
0.64  419 
0.64  360 
0.64  302 


0.64  243 


0.64  185 
0.64  127 
0.64  069 
0.64  011 
0.63  953 
0.63  895 
0.63  837 
0.63  779 
0.63  721 


0.63  664 


L.  Tan.   L.  Sin 


9.99  040 


9.99  038 
9.99  035 
9.99  032 
9.99  030 
9.99  027 
9.99  024 
9.99  022 
9.99  019 
9.99  016 


9.99  013 


9.99  011 
9.99  008 
9.99  005 
9.99  002 
9.99  000 
9.98  997 
9.98  994 
9.98  991 
9.98  989 


9.98  986 


9.98  983 
9.98  980 
9.98  978 
9.98  975 
9.98  972 
9.98  969 
9.98  967 
9.98  964 
9.98  961 


9.98  958 


9.98  955 
9.98  953 
9.98  950 
9.98  947 
9.98  944 
9.98  941 
9.98  938 
9.98  936 
9.98  933 


9.98  930 


9.98  927 
9.98  924 
9.98  921 
9.98  919 
9.98  916 
9.98  913 
9.98  910 
9.98  907 
9.98  904 


9.98  901 


9.98  898 
9.98  896 
9.98  893 
9.98  890 
9.98  887 
9.98  884 
9.98  881 
9.98  878 
9.98  875 


9.98  872 


30 


P.P. 


63  I  62  I  61 


5  5-2 
10  lo.s 
15,15.8 
20  21.0 
2526. 2 
3o!3i-5 
35!36.8 
40  42.0 
4S'47-2 
SO  52.S 

ss  57.8 


1  5-2  5-1 
10.3  10.2 
jiS-5  15-2 
20.7  20.3 


25.8  25.4 
I31.030.5 
36.2  35.6 
41-340.7 
46.545.8 
51.7  50.8 
'56.8I55.9 


// 

60 

69  1  58 

5 

5.0!  4.9!  4.8 

10 

10.0:   9.8    9.7 

15  15-0  I4.8|I4.S 

20  20.0  19.7  19.3 

25  25.o'24.6  24.2 

30  30.0  29. s  29.0 

35,35.034-433.8 

40|40.o 

39.338.7 

4545.0 

44.2  43.5 

50*50.0 

49.2  48.3 

55 

S5.0 

54.1  53.2 

57  I 
4.8 
9.5 
15  14.2 
20  19.0 
2523.8: 
3028.5 
35  33.2 
4038.0 
4542.8 
5047.5 
5552.2 


56 

4.7 
9-3 
14.0 


55 

4.6 
9.2 
13.8 


18.7  18.3 
23.3,22.9 
28.0  27.5 
32.7132.1 
37.3{36.7 
42.0'41.2 

46.7145.8 
5l.3'S0.4 


"13  12 
5  0.2  0.2 
100.5  0.3 
i5lo.8'o.s 
20:1.0  0.7 
25J1.2  0.8 
30  1.51.0 
3S'l.8;i.2 

40  2.0  1.3 

45,2.2  1.5 

502.5 1.7 

552.8  1.8 


P.P. 


77° 


13° 


137 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
J9 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


9.35  209 


9.35  263 
9.35  318 
9.35  373 
9.35  427 
9.35  481 
9.35  536 
9.35  590 
9.35  644 
9.35  698 


9.35  752 


9.35  806 
9.35  860 
9.35  914 

9.35  968 

9.36  022 
9.36  075 
9.36  129 
9.36  182 
9.36  236 


9.36  289 


9.36  342 
9.36  395 
9.36  449 
9.36  502 
9.36  555 
9.36  608 
9.36  660 
9.36  713 
9.36  766 


9.36  819 


9.36  871 
9.36  924 

9.36  976 

9.37  028 
9.37  081 
9.37  133 
9.37  185 
9.37  237 
9.37  289 


9.37  341 


9.37  393 
9.37  445 
9.37  497 
9.37  549 
9.37  600 
9.37  652 
9.37  703 
9.37  755 
9.37  806 


37  858 


37  909 

37  960 

38  011 
38  062 
38  113 
38  164 
38  215 
38  266 
38  317 


38  368 


L.  Cos. 


54 
55 
55 
54 
54 
55 
54 
54 
54 
54 
54 
54 
54 
54 
54 
53 
54 
53 
54 
53 
53 
53 
54 
53 
53 
53 
52 
53 
53 
53 
52 
53 
52 
52 
53 
52 
52 
52 
52 
52 
52 
52 
52 
52 
51 
52 
51 
52 
51 
52 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 


9.36  336 


9.36 
9.36 
9.36 
9.36 
9.36 
9.36 
9.36 
9.36 
9.36 


394 

452 
509 
566 
624 
681 
738 
795 
852 


9.36  909 


9.36 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 


966 
023 
080 
137 
193 
250 
306 
363 
419 


9.37  476 


9.37 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 
9.37 


532 
588 
644 
700 
756 
812 
868 
924 
980 


9.38  035 


9.38 
9.38 
9.38 
9.38 
9.38 
9.38 
9.38 
9.38 
9.38 


091 
147 
202 
257 
313 
368 
423 
479 
534 


9.38  589 


9.38 
9.38 
9.38 
9.38 
9.38 
9.38 
9.38 
9.39 
9.39 
9.39 


644 
699 
754 
808 
863 
918 
972 
027 
082 


136 


9.39 
9.39 
9.39 
9.39 
9.39 
9.39 
9.39 
9.39 
9.39 


190 
245 
299 
353 
407 
461 
515 
569 
623 


9.39  677 


d.   L.  Cot.  c.d. 


58 
58 
57 
57 
58 
57 
57 
57 
57 
57 
57 
57 
57 
57 
56 
57 
56 
57 
56 
57 
56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 
56 
55 
55 
56 
55 
55 
56 
55 
55 
55 
55 
55 
54 
55 
55 
54 
55 
55 
54 
54 
55 
54 
54 
54 
54 
54 
54 
54 
54 


0.63  664 


0.63  606 
0.63  548 
0.63  491 
0.63  434 
0.63  376 
0.63  319 
0.63  262 
0.63  205 
0.63  148 


0.63  091 


0.63  034 
0.62  977 
0.62  920 
0.62  863 
0.62  807 
0.62  750 
0.62  694 
0.62  637 
0.62  581 


0.62  524 


0.62  468 
0.62  412 
0.62  356 
0.62  300 
0.62  244 
0.62  188 
0.62  132 
0.62  076 
0.62  020 


0.61  965 


0.61  909 
0.61  853 
0.61  798 
0.61  743 
0.61  687 
0.61  632 
0.61  577 
0.61  521 
0.61  466 


0.61411 


0.61  356 
0.61  301 
0.61  246 
0.61  192 
0.61  137 
0.61  082 
0.61  028 
0.60  973 
0.60  918 


0.60  864 


0.60  810 
0.60  755 
0.60  701 
0.60  647 
0.60  593 
0.60  539 
0.60  485 
0.60  431 
0.60  377 


0.60  323 


L.  Tan.   L.  Sin 


9.98  872 


9.98  869 
9.98  867 
9.98  864 
9.98  861 
9.98  858 
9.98  855 
9.98  852 
9.98  849 
9.98  846 


9.98  843 


9.98  840 
9.98  837 
9.98  834 
9.98  831 
9.98  828 
9.98  825 
9.98  822 
9.98  819 
9.98  816 


9.98  813 


9.98  810 
9.98  807 
9.98  804 
9.98  801 
9.98  798 
9.98  795 
9.98  792 
9.98  789 
9.98  786 


9.98  783 


9.98  780 
9.98  777 
9.98  774 
9.98  771 
9.98  768 
9.98  765 
9.98  762 
9.98  759 
9.98  756 


9.98  753 


9.98  750 
9.98  746 
9.98  743 
9.98  740 
9.98  737 
9.98  734 
9.98  731 
9.98  728 
9.98  725 


9.98  722 


9.98  719 
9.98  715 
9.98  712 
9.98  709 
9.98  706 
9.98  703 
9.98  700 
9.98  697 
9.98  694 


9.98  690 


P.P. 


n 

51  4-8; 

lo;  9-7| 

IS  14-5 
20  19.3 

25i24.2j 
30  29.0' 
35133-8 
4038.71 
45  43-5; 
5048-3 
55  S3-2I 


58  I  57  I  56 
4.8  4-7 
9-5  9-3 
14.2  14.0 
19.0  18.7 
23-8  23.3 
28. S  28.0 
33-2  32.7 
38.037.3 
42.8  42.0 
47-5  46.7 
52.2  S1.3 


55  I  54     53 


4-6 
9-2 
13-8 
18.3 


25122.9 
30  27.5 
35  32.1 
4036.7 
45  41-2 
50I45.8 
55 150.4 


4-5    4-4 

9.0 
13-5  13-2 
18.0  17-7 
22.5  22.1 
27.0  26.5 
31-5  30.9 
36.0  35-3 
40.5  39-8 
45-0,44-2 
49.5'48.6 


52     51 


4-3 

8.7 

13.0 

17-3 

21.7 


30  26.0 
35*30.3 
40134.7 
45j39-0 
5043-3 
SS!47.7 


4.2 
8-5 
12.8 
17.0 
21.2 
25-5 
29.8 
340 
38.2 
42-5 
46.8 


4     3     2 


50.3 
100.7 
15  i-o 
20  1.3 
25'i-7 


i.o  0.7 
1.20.8 


302.0  1.5  1.0 
35  2.3  i.8|i.2 

40  2.712.0  1.3 
45i3-0,2.2  i.S 
50  3-3  2.5  1.7 
55  3-7  2.8  1.8 


P.P. 


7AO 


138 


14^ 


10 


21 
22 
23 
24 
25 
26 
27 
28 
29 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


51 

52 
53 
54 

55 
56 

57 
58 

60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d 


9.38  368 


9.38  418 
9.38  469 
9.38  519 
9.38  570 
9.38  620 
9.38  670 
9.38  721 
9.38  771 
9.38  821 


9.38  871 


9.38  921 

9.38  971 

9.39  021 
9.39  071 
9.39  121 
9.39  170 
9.39  220 
9.39  270 
9.39  319 


9.39  369 


9.39  418 
9.39  467 
9.39  517 
9.39  566 
9.39  615 
9.39  664 
9.39  713 
9.39  762 
9.39  811 


9.39  860 


9.39  909 

9.39  958 

9.40  006 
9.40  055 
9.40  103 
9.40  152 
9.40  200 
9.40  249 
9.40  297 


9.40  346 


9.40  394 
9.40  442 
9.40  490 
9.40  538 
9.40  586 
9.40  634 
9.40  682 
9.40  730 
9.40  778 


9.40  825 


9.40  873 
9.40  921 

9.40  968 

9.41  016 
9.41  063 
9.41  111 
9.41  158 
9.41  205 
9.41  252 


9.41  300 


50 
51 
50 
51 
50 
50 
51 
50 
50 
50 
50 
50 
50 
50 
50 
49 
50 
50 
49 
50 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 
49 
48 
49 
48 
49 
48 
49 
48 
48 
48 
48 
48 
48 
48 
48 
48 
47 
48 
48 
47 
48 
47 
48 
47 
47 
47 
48 


9.39  677 


9.39  731 
9.39  785 
9.39  838 
9.39  892 
9.39  945 

9.39  999 

9.40  052 
9.40  106 
9.40  159 


9.40  212 


9.40  266 
9.40  319 
9.40  372 
9.40  425 
9.40  478 
9.40  531 
9.40  584 
9.40  636 
9.40  689 


9.40  742 


9.40  795 
9.40  847 
9.40  900 

9.40  952 

9.41  005 
9.41  057 
9.41  109 
9.41  161 
9.41  214 


9.41  266 


9.41  318 
9.41  370 
9.41  422 
9.41  474 
9.41  526 
9.41  578 
9.41  629 
9.41  681 
9.41  733 


9.41  784 


9.41  836 
9.41  887 
9.41  939 

9.41  990 

9.42  041 
9.42  093 
9.42  144 
9.42  195 
9.42  246 


9.42  297 


L.  Cos.   d.   L.  Cot. 


9.42  348 
9.42  399 
9.42  450 
9.42  501 
9.42  552 
9.42  603 
9.42  653 
9.42  704 
9.42  755 


9.42  805 


54 
54 
53 
54 
53 
54 
53 
54 
53 
53 
54 
53 
53 
53 
53 
53 
53 
52 
53 
53 
53 
52 
53 
52 
53 
52 
52 
52 
53 
52 
52 
52 
52 
52 
52 
52 
51 
52 
52 
51 
52 
51 
52 
51 
51 
52 
51 
51 
51 
51 
51 
51 
51 
51 
51 
51 
50 
51 
51 
50 


c.d. 


0.60  323 
0^60  269 
0.60  215 
0.60  162 
0.60  108 
0.60  055 
0.60  001 
0.59  948 
0.59  894 
0.59841 
0.59  788 


0.59  734 
0.59  681 
0.59  628 
0.59  575 
0.59  522 
0.59  469 
0.59  416 
0.59  364 
0.59  311 


0.59  258 


0.59  205 
0.59  153 
0.59  100 
0.59  048 
0.58  995 
0.58  943 
0.58  891 
0.58  839 
0.58  786 


0.58  734 
0.58  682" 
0.58  630 
0.58  578 
0.58  526 
0.58  474 
0.58  422 
0.58  371 
0.58  319 
0.58  267 


0.58  216 


0.58  164 
0.58  113 
0.58  061 
0.58  010 
0.57  959 
0.57  907 
0.57  856 
0.57  805 
0.57  754 


0.57  703 


0.57  652 
0.57  601 
0.57  550 
0.57  499 
0.57  448 
0.57  397 
0.57  347 
0.57  296 
0.57  245 


0.57  195 


9.98  690 


9.98  687 
9.98  684 
9.98  681 
9.98  678 
9.98  675 
9.98  671 
9.98  668 
9.98  665 
9.98  662 


9.98  659 


9.98  656 
9.98  652 
9.98  649 
9.98  646 
9.98  643 
9.98  640 
9.98  636 
9.98  633 
9.98  630 


9.98  627 


9.98  623 
9.98  620 
9.98  617 
9.98  614 
9.98  610 
9.98  607 
9.98  604 
9.98  601 
9.98  597 


9.98  594 


9.98  591 
9.98  588 
9.98  584 
9.98  581 
9.98  578 
9.98  574 
9.98  571 
9.98  568 
9.98  565 


9.98  561 


9.98  558 
9.98  555 
9.98  551 
9.98  548 
9.98  545 
9.98  541 
9.98  538 
9.98  535 
9.98  531 


9.98  528 


9.98  525 
9.98  521 
9.98  518 
9.98  515 
9.98  511 
9.98  508 
9.98  505 
9.98  501 
9.98  498 


9.98  494 


L.  Tan. 

75° 


L.  Sin. 


d. 


60 

59 

58 
57 
56 
55 
54 
53 
52 

50^ 

49 

48 
47 
46 
45 
44 
43 
42 

40 

39 
38 
37 
36 
35 
34 
33 
31 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 

10 

9 

8 
7 
6 
5 
4 
3 
2 
1 
0 


P.P. 


54 


52 


5  4-5  4- 
10 j  Q.o!  8. 
I5|l3.5ll3. 
20|i8.oli7, 

25J22.S!22, 

3o'27.o  26, 
3S'3i.5  30. 

40  36.0  35. 

4540. S39> 
50'4S.o:44. 
SS;49.5i48. 


4.3 
8.7 

2  j  13.0 

7  17.3 

1  21.7 
S'26.0 
930.3 

3  34-7 
839.0 

2  43.3 
647.7 


"I  51 

5  4-2 
io|  8.5 
IS  12.8 
20  17.0 

2S[2I.2 
30:25.5 
35,29.8 
4034-0 
4538.2 
5042.5 
5546.8 


50  I  49 

4.21  4.1 

8.3    8.2 

12.5  12.2 

16.7  16.3 

20.8  20.4 
25.0  24.S 


29.2 
33.3 
37.5 
41.7 
45.8 


28.6 
32.7 
36.8 
40.8 
44.9 


48  I  47 


4-0    3-9 

8.0    7.8 

i2.o'ii.8 

16.0,15.7 

20.0|l9.6 

24.0  23.5 
35]28.o  27.4 
4032.031.3 
4536.035.2 
5o'40.o!39.2 
55  44.o'43.i 


"  4  I  3 
5  o.3'o.2 
10  0.7  0.5 
15  1.0^0.8 
20  1.311.0 

25  I.7|I.2 
30  2.0jl.5 
35  2.3I1.8 
40  2.7|2.0 
453.0  2.2 
50  3.3  2.5 
55  3.7  2.8 


P.P. 


15° 


139 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


60 


L.  Sin. 


9.41  300 


9.41  347 
9.41  394 
9.41  441 
9.41  488 
9.41  535 
9.41  582 
9.41  628 
9.41  675 
9.41  722 


9.41  768 


9.41815 
9.41  861 
9.41  908 

9.41  954 

9.42  001 
9.42  047 
9.42  093 
9.42  140 
9.42  186 


9.42  232 


9.42  278 
9.42  324 
9.42  370 
9.42  416 
9.42  461 
9.42  507 
9.42  553 
9.42  599 
9.42  644 


9.42  690 


9.42  735 
9.42  781 
9.42  826 
9.42  872 
9.42  917 

9.42  962 

9.43  008 
9.43  053 
9.43  098 


9.43  143 


9.43  188 
9.43  233 
9.43  278 
9.43  323 
9.43  367 
9.43  412 
9.43  457 
9.43  502 
9.43  546 


9.43  591 


9.43  635 
9.43  680 
9.43  724 
9.43  769 
9.43  813 
9.43  857 
9.43  901 
9.43  946 
9.43  990 


9.44  034 


L.  Cos. 


d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


47 
47 
47 
47 
47 
47 
46 
47 
47 
46 
47 
46 
47 
46 
47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
45 
46 
46 
46 
45 
46 
45 
46 
45 
46 
45 
45 
46 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 


42  805 


,42  856 
,42  906 
,42  957 
43  007 
43  057 
43  108 
43  158 
43  208 
43  258 


43  308 


43  358 
43  408 
43  458 
43  508 
43  558 
43  607 
43  657 
43  707 
43  756 


43  806 


43  855 
43  905 

43  954 

44  004 
44  053 
44  102 
44  151 
44  201 
44  250 


44  299 


44  348 
44  397 
44  446 
44  495 
44  544 
44  592 
44  641 
44  690 
44  738 


44  787 


44  836 
44  884 
44  933 

44  981 

45  029 
45  078 
45  126 
45  174 
45  222 


45  271 


45  319 
45  367 
45  415 
45  463 
45  511 
45  559 
45  606 
45  654 
45  702 


45  750 


d.   L.  Cot.  c.d 


51 
50 
51 
50 
50 
51 
50 
50 
50 
50 
50 
50 
50 
50 
50 
49 
50 
50 
49 
50 
49 
50 
49 
50 
49 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 
48 
49 
59 
48 
49 
48 
48 
49 
48 
48 
48 
49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 


0.57  195 


0.57  144 
0.57  094 
0.57  043 
0.56  993 
0.56  943 
0.56  892 
0.56  842 
0.56  792 
0.56  742 


0.56  692 


0.56  642 
0.56  592 
0.56  542 
0.56  492 
0.56  442 
0.56  393 
0.56  343 
0.56  293 
0.56  244 


0.56  194 


0.56  145 
0.56  095 
0.56  046 
0.55  996 
0.55  947 
0.55  898 
0.55  849 
0.55  799 
0.55  750 


0.55  701 


0.55  652 
0.55  603 
0.55  554 
0.55  505 
0.55  456 
0.55  408 
0.55  359 
0.55  310 
0.55  262 


0.55  213 


0.55  164 
0.55  116 
0.55  067 
0.55  019 
0.54  971 
0.54  922 
0.54  874 
0.54  826 
0.54  778 


0.54  729 


0.54  681 
0.54  633 
0.54  585 
0.54  537 
0.54  489 
0.54  441 
0.54  394 
0.54  346 
0.54  298 


0.54  250 


L.  Tan.   L.  Sin. 


9.98  494 


9.98  491 
9.98  488 
9.98  484 
9.98  481 
9.98  477 
9.98  474 
9.98  471 
9.98  467 
9.98  464 


9.98  460 


9.98  457 
9.98  453 
9.98  450 
9.98  447 
9.98  443 
9.98  440 
9.98  436 
9.98  433 
9.98  429 


9.98  426 


9.98  422 
9.98  419 
9.98  415 
9.98  412 
9.98  409 
9.98  405 
9.98  402 
9.98  398 
9.98  395 


9.98  391 


9.98  388 
9.98  384 
9.98  381 
9.98  377 
9.98  373 
9.98  370 
9.98  366 
9.98  363 
9.98  359 


9.98  356 


9.98  352 
9.98  349 
9.98  345 
9.98  342 
9.98  338 
9.98  334 
9.98  331 
9.98  327 
9.98  324 


9.98  320 


9.98  317 
9.98  313 
9.98  309 
9.98  306 
9.98  302 
9.98  299 
9.98  295 
9.98  291 
9.98  288 


9.98  284 


d. 


40 


P.P. 


I 

S  4-2 
lo!  8.5 
I5;i2.8 

20|I7.0 
25  21.2 

3025.5 

35  29.8 

4034-0 
45:38.2 

SOj42.5 
5SI46.8, 


51  '  50  I  49 


4.2  4.1 

8.3  8.2 
12.5  12.2 

16.7  16.3 

20.8  20.4 
25.0I24.S 
29.2  28.6 
33.332.7 
37.536.8 
41.7  40.8 
45.844.9 


48  I  47 


5\  4-0 
10  8.0 
IS1I2.0 
20J16.0 
25  j  20.0 
30124.0 
35  28.0 
40  32.0 
45  36.0 
SO  40.0 
55  44-0 


3.9  3.8 
7.8;  7.7 
11.8  11.5 
15.715.3 
19.6  19.2 
23.5I23.0 
27.426.8 


31-3 
35.2 
39-2 
43.1 


30.7 
34-5 
38.3 
42.2 


45     44 


3.8 

7.5 

II. 2 


3.7 
7.3 

II.O 


15.0  14.7 
18.8  18.3 
22.5  22.0 
26.2  25.7 

30.0;29.3 
33.8|33.0 
37.5  36.7 


55141.2140.3 


50.3 
10  0.7 
15  i.o 
20J1.3 

25  1.7 

30i2.0 
35|2.3 
40  2.7  2.0 
45  3.02.2 
50  3.3  2.5 
S5I3.7  2.8 


P.P. 


140 


16< 


10 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 
52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin. 


9.44  034 


9.44  078 
9.44  122 
9.44  166 
9.44  210 
9.44  253 
9.44  297 
9.44  341 
9.44  385 
9.44  428 


9.44  472 


9.44  516 
9.44  559 
9.44  602 
9.44  646 
9.44  689 
9.44  733 
9.44  776 
9.44  819 
9.44  862 


9.44  905 


9.44  948 

8.44  992 

9.45  035 
9.45  077 
9.45  120 
9.45  163 
9.45  206 
9.45  249 
9.45  292 


9.45  334 


9.45  377 
9.45  419 
9.45  462 
9.45  504 
9.45  547 
9.45  589 
9.45  632 
9.45  674 
9.45  716 


9.45  758 


9.45  801 
9.45  843 
9.45  885 
9.45  927 

9.45  969 

9.46  011 
9.46  053 
9.46  095 
9.46  136 


9.46  178 


9.46  220 
9.46  262 
9.46  303 
9.46  345 
9.46  386 
9.46  428 
9.46  469 
9.46  511 
9.46  552 


9.46  594 
L.  Cos. 


44 

44 

44 

44 

43 

44 

44 

44 

43 

44 

44 

43 

43 

44 

43 

44 

43 

43 

43 

43 

43 

44 

43 

42 

43 

43 

43 

43 

43 

42 

43 

42 

43 

42 

43 

42 

43 

42 

42 

42 

43 

42 

42 

42 

42 

42 

42 

42 

41 

42 

42 

42 

41 

42 

41 

42 

41 

42 

41 

42 


L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


9.45  750 


9.45  797 
9.45  845 
9,45  892 
9.45  940 

9.45  987 

9.46  035 
9.46  082 
9.46  130 
9.46  177 


9.46  224 


9.46  271 
9.46  319 
9.46  366 
9.46  413 
9.46  460 
9.46  507 
9.46  554 
9.46  601 
9.46  648 


9.46  694 


9.46  741 
9.46  788 
9.46  835 
9.46  881 
9.46  928 

9.46  975 

9.47  021 
9.47  068 
9.47  114 


9.47  160 


47 
48 
47 
48 
47 
48 
47 
48 
47 
47 
47 
48 
47 
47 
47 
47 
47 
47 
47 
46 
47 
47 
47 
46 
47 
47 
46 
47 
46 
46 
47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
45 
46 
46 
46 
45 
46 
45 
46 
45 
45 
46 
45 
45 
46 

4S 
9.48  534 

L.  Cot.   c.d 


9.47  207 
9.47  253 
9.47  299 
9.47  346 
9.47  392 
9.47  438 
9.47  484 
9.47  530 
9.47  576 


9.47  622 


9.47  668 
9.47  714 
9.47  760 
9.47  806 
9.47  852 
9.47  897 
9.47  943 

9.47  989 

9.48  035 


9.48  080 


9.48  126 
9.48  171 
9.48  217 
9.48  262 
9.48  307 
9.48  353 
9.48  398 
9.48  443 
9.48  489 


0.54  250 


0.54  203 
0.54  155 
0.54  108 
0.54  060 
0.54  013 
0.53  965 
0.53  918 
0.53  870 
0.53  823 


0.53  776 


0.53  729 
0.53  681 
0.53  634 
0.53  587 
0.53  540 
0.53  493 
0.53  446 
0.53  399 
0.53  352 


0.53  306 


0.53  259 
0.53  212 
0.53  165 
0.53  119 
0.53  072 
0.53  025 
0.52  979 
0.52  932 
0.52  886 


0.52  840 


0.52  793 
0.52  747 
0.52  701 
0.52  654 
0.52  608 
0.52  562 
0.52  516 
0.52  470 
0.52  424 


0.52  378 


0.52  332 
0.52  286 
0.52  240 
0.52  194 
0.52  148 
0.52  103 
0.52  057 
0.52  011 
0.51  965 
0.51  920 


0.51  874 
0.51  829 
0.51  783 
0.51  738 
0.51  693 
0.51  647 
0.51  602 
0.51  557 
0.51  511 


0.51  466 


9.98  284 


9.98  281 
9.98  277 
9.98  273 
9.98  270 
9.98  266 
9.98  262 
9.98  259 
9.98  255 
9.98  251 


9.98  248 


9.98  244 
9.98  240 
9.98  237 
9.98  233 
9.98  229 
9.98  226 
9.98  222 
9.98  218 
9.98  215 


9.98  211 


9.98  207 
9.98  204 
9.98  200 
9.98  196 
9.98  192 
9.98  189 
9.98  185 
9.98  181 
9.98  177 


9.98  174 


9.98  170 
9.98  166 
9.98  162 
9.98  159 
9.98  155 
9.98  151 
9.98  147 
9.98  144 
9.98  140 


9.98  136 


9.98  132 
9.98  129 
9.98  125 
9.98  121 
9.98  117 
9.98  113 
9.98  110 
9.98  106 
9.98  102 
9.98  098 
9.98  094 
9.98  090 
9.98  087 
9.98  083 
9.98  079 
9.98  075 
9.98  071 
9.98  067 
9.98  063 


9.98  060 


L.  Tan.      L.  Sin.      d. 


P.P. 


" 

48 

47 

46 

5 

4.0 

3.9 

3.8 

10 

8.0 

7.8 

7.7 

15 

12.0 

11.8 

1 1.5 

20 

i6.o!i5.7 

15.3 

25 

20.0 

19.6 

19.2 

30 

24.0 

23.5(23.0 

35  28.0 

27.4  26.8 

40  32.0 

31.330.7 

45  36.0 

35.234.5 

SO  40.0 

39.238.3 

55 

44.0 

43.1 

42.2 

45     44     43 


3.8 

7.5 

II. 2 

15-0 

18.8 


3022. 5 
35  26.2 
40!  30.0 
45  33.8 
5037.5 
S5I41.2 


3-7 

7.3 

II.O 

14.7 
18.3 
22.0 
25-7 
29-3 
33-0 
136.7 
I40.3 


3.6 
7.2 
10.8 
14.3 
17.9 
2I.S 
25.1 
28.7 
32.2 
35.8 
39.4 


"I  42 

5  3.5 
10  7.0 
15  10.5 
20  14.0 
17.5 
21.0 


41 

3.4 
6.8 
10.2 
13.7 
17. 1 

20.5 


24.5123.9 

28.0|27.3 

31.530.8 
35.034-2 
38.537.6 


"  ;  4  I  3 

5  o.3'o.2 
10  0.7  0.5 
IS  i.ojo.S 
20  1.3  i.o 
25  1.7  1.2 
30  2.0I1.S 
35  2.3  1.8 
40  2.7  2.0 
45  3.0  2.2 
50  3-3  2.5 
S53.7I2.8 


P.P. 


17* 


141 


L.  Sin. 


L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


P.P. 


1 

2 

3 

4 

5 

6 

7 

8 

_9_ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


60 


9.46  594 


9.46  635 
9.46  676 
9.46  717 
9.46  758 
9.46  800 
9.46  841 
9.46  882 
9.46  923 
9.46  964 


9.47  005 


9.47  045 
9.47  086 
9.47  127 
9.47  168 
9.47  209 
9.47  249 
9.47  290 
9.47  330 
9.47  371 


9.47  411 


9.47  452 
9.47  492 
9.47  533 
9.47  573 
9.47  613 
9.47  654 
9.47  694 
9.47  734 
9.47  774 


9.47  814 


9.47  854 
9.47  894 
9.47  934 

9.47  974 

9.48  014 
9.48  054 
9.48  094 
9.48  133 
9.48  173 


9.48  213 


9.48  252 
9.48  292 
9.48  332 
9.48  371 
9.48  411 
9.48  450 
9.48  490 
9.48  529 
9.48  568 


9.48  607 


9.48  647 
9.48  686 
9.48  725 
9.48  764 
9.48  803 
9.48  842 
9.48  881 
9.48  920 
9.48  959 


9.48  998 


41 
41 
41 
41 
42 
41 
41 
41 
41 
41 
40 
41 
41 
41 
41 
40 
41 
40 
41 
40 
41 
40 
41 
40 
40 
41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
39 
40 
40 
39 
40 
40 
39 
40 
39 
40 
39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 


48  534 


48  579 
48  624 
48  669 
48  714 
48  759 
48  804 
48  849 
48  894 
48  939 


48  984 


49  029 
49  073 
49  118 
49  163 
49  207 
49  252 
49  296 
49  341 
49  385 


49  430 


49  474 
49  519 
49  563 
49  607 
49  652 
49  696 
49  740 
49  784 
49  828 


49  872 


49  916 

49  960 

50  004 
50  048 
50  092 
50  136 
50  180 
50  223 
50  267 


50  311 


50  355 
50  398 
50  442 
50  485 
50  529 
50  572 
50  616 
50  659 
50  703 


50  746 


50  789 
50  833 
50  876 
50  919 

50  962 
51005 
51048 
51092 

51  135 


51  178 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
43 
44 
43 
44 
43 
44 
43 
44 
43 
43 
44 
43 
43 
43 
43 
43 
44 
43 
43 


0.51  466 


9.98  060 


0.51421 
0.51  376 
0.51  331 
0.51  286 
0.51  241 
0.51  196 
0.51  151 
0.51  106 
0.51  061 
0.51  016 
0.50  971 
0.50  927 
0.50  882 
0.50  837 
0.50  793 
0.50  748 
0.50  704 
0.50  659 
0.50  615 


9.98  056 
9.98  052 
9.98  048 
9.98  044 
9.98  040 
9.98  036 
9.98  032 
9.98  029 
9.98  025 


9.98  021 


9.98  017 
9.98  013 
9.98  009 
9.98  005 
9.98  001 
9.97  997 
9.97  993 
9.97  989 
9.97  986 


0.50  570 


9.97  982 


0.50  526 
0.50  481 
0.50  437 
0.50  393 
0.50  348 
0.50  304 
0.50  260 
0.50  216 
0.50  172 


9.97  978 
9.97  974 
9.97  970 
9.97  966 
9.97  962 
9.97  958 
9.97  954 
9.97  950 
9.97  946 


0.50  128 


9.97  942 


0.50  084 
0.50  040 
0.49  996 
0.49  952 
0.49  908 
0.49  864 
0.49  820 
0.49  777 
0.49  733 


9.97  938 
9.97  934 
9.97  930 
9.97  926 
9.97  922 
9.97  918 
9.97  914 
9.97  910 
9.97  906 


049^89 
0.49  645 
0.49  602 
0.49  558 
0.49  515 
0.49  471 
0.49  428 
0.49  384 
0.49  341 
0.49  297 


9.97  902 


9.97  898 
9.97  894 
9.97  890 
9.97  886 
9.97  882 
9.97  878 
9.97  874 
9.97  870 
9.97  866 


0.49  254 


9.97  861 


0.49  211 
0.49  167 
0.49  124 
0.49  081 
0.49  038 
0.48  995 
0.48  952 
0.48  908 
0.48  865 


9.97  857 
9.97  853 
9.97  849 
9.97  845 
9.97  841 
9.97  837 
9.97  833 
9.97  829 
9.97  825 


0.48  822 


9.97  821 


10 


45     44     43 


3.8 
7-5 

II. 2 

15.0 
i8.8 

22.5 

26.2 


40  30.0 
4533-8 
5037.S 
5541.2 


3.7  3.6 
7-3  7.2 
ii.o  10.8 
I4.7ji4.3 
18.3  17-9 
22.0  21.5 
25.7  25. 1 
29.3'28.7 
33.032.2 
36.7'35.8 
40.3:39.4 


42     41 


51  3.5 
10  7.0 
15  10.5 
20' 14.0 

25;I7.S 
3021.0 


3.4 

6.8 
10.2 
13.7 
17.1 

20.5 


35  24.S  23.9 


40  28.0 
45  31.5 
SO  35.0 
5538.5 


27.3 
30.8 
34-2 
37.6 


"  I  40  I  39 

5  3.3  3.2 
io[  6.7  6.5 
15  lO.O    9.8 

20  13.3;  13.0 
25:16.7  16.2 


30  20.0 

35  23.3 

40;  26.7 
45  30.0 
50  33-3 
55  36.7 


19.S 
22.8 
26.0 
29.2 
32.5 
35.8 


"1 5 1 

50.4! 
10  0.8 
15  1.2 
20  1.7 
252. 1 
30J2.5 
35  2.9 


4     3 


2.7  2.0 
3.0:2.2 

3.3  2.S 

3.7i2.8 


L.  Cos. 


d.       L.  Cot.     c.d 


L.  Tan.      L.  Sin. 


d. 


P.P. 


142 


20 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 

42 
43 
44 
45 
46 
47 
48 
49 


50 


51 

52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


9.48  998 


9.49  037 
9.49  076 
9.49  115 
9.49  153 
9.49  192 
9.49  231 
9.49  269 
9.49  308 
9.49  347 


9.49  385 


9.49  424 
9.49  462 
9.49  500 
9.49  539 
9.49  577 
9.49  615 
9.49  654 
9.49  692 
9.49  730 


9.49  768 


9.49  806 
9.49  844 
9.49  882 
9.49  920 
9.49  958 

9.49  996 

9.50  034 
9.50  072 
9.50  110 


9.50  148 


9.50  185 
9.50  223 
9.50  261 
9.50  298 
9.50  336 
9.50  374 
9.50  411 
9.50  449 
9.50  486 


9.50  523 


9.50  561 
9.50  598 
9.50  635 
9.50  673 
9.50  710 
9.50  747 
9.50  784 
9.50  821 
9.50  858 


9.50  896 


50  933 

50  970 
51007 
51043 
51080 

51  117 
51  154 
51  191 
51  227 


51  264 


39 
39 
39 

38 
39 
39 
38 
39 
39 
38 
39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
37 
38 
38 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
38 
37 
37 
37 
37 
37 
38 
37 
37 
37 
36 
37 
37 
37 
37 
36 
37 


18^ 
L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


9.51  178 


9.51  221 
9.51  264 
9.51  306 
9.51  349 
9.51  392 
9.51  435 
9.51  478 
9.51  520 
9.51  563 


9.51  606 


9.51  648 
9.51  691 
9.51  734 
9.51  776 
9.51  819 
9.51  861 
9.51  903 
9.51  946 
9.51  988 


9.52  031 


9.52  073 
9.52  115 
9.52  157 
9.52  200 
9.52  242 
9.52  284 
9.52  326 
9.52  368 
9.52  410 


9.52  452 


9.52  494 
9.52  536 
9.52  578 
9.52  620 
9.52  661 
9.52  703 
9.52  745 
9.52  787 
9.52  829 


9.52  870 


9.52  912 
9.52  953 

9.52  995 

9.53  037 
9.53  078 
9.53  120 
9.53  161 
9.53  202 
9.53  244 


9.53  285 


0.53  327 
9.53  368 
9.53  409 
9.53  450 
9.53  492 
9.53  533 
9.53  574 
9.53  615 
9.53  656 


9.53  697 


43 

43 

42 

43 

43 

43 

43 

42 

43 

43 

42 

43 

43 

42 

43 

42 

42 

43 

42 

43 

42 

42 

42 

43 

42 

42 

42 

42 

42 

42 

42 

42 

42 

42 

41 

42 

42 

42 

42 

41 

42 

41 

42 

42 

41 

42 

41 

41 

42 

41 

42 

41 

41 

41 

42 

41 

41 

41 

41 

41 


0.48  822 


0.48  779 
0.48  736 
0.48  694 
0.48  651 
0.48  608 
0.48  565 
0.48  522 
0.48  480 
0.48  437 


0.48  394 


0.48  352 
0.48  309 
0.48  266 
0.48  224 
0.48  181 
0.48  139 
0.48  097 
0.48  054 
0.48  012 


0.47  969 


0.47  927 
0.47  885 
0.47  843 
0.47  800 
0.47  758 
0.47  716 
0.47  674 
0.47  632 
0.47  590 


9.97  821 


9.97  817 
9.97  812 
9.97  808 
9.97  804 
9.97  800 
9.97  796 
9.97  792 
9.97  788 
9.97  784 


9.97  779 


9.97  775 
9.97  771 
9.97  767 
9.97  763 
9.97  759 
9.97  754 
9.97  750 
9.97  746 
9.97  742 


9.97  738 


0.47  548 


0.47  506 
0.47  464 
0.47  422 
0.47  380 
0.47  339 
0.47  297 
0.47  255 
0.47  213 
0.47  171 


0.47  130 


0.47  088 
0.47  047 
0.47  005 
0.46  963 
0.46  922 
0.46  880 
0.46  839 
0.46  798 
0.46  756 


0.46  715 


0.46  673 
0.46  632 
0.46  591 
0.46  550 
0.46  508 
0.46  467 
0.46  426 
0.46  385 
0.46  344 


0.46  303 


9.97  734 
9.97  729 
9.97  725 
9.97  721 
9.97  717 
9.97  713 
9.97  708 
9.97  704 
9.97  700 


9.97  696 


9.97  691 
9.97  687 
9.97  683 
9.97  679 
9.97  674 
9.97  670 
9.97  666 
9.97  662 
9.97  657 


9.97  653 


9.97  649 
9.97  645 
9.97  640 
9.97  636 
9.97  632 
9.97  628 
9.97  623 
9.97  619 
9.97  615 


9.97  610 


9.97  606 
9.97  602 
9.97  597 
9.97  593 
9.97  589 
9.97  584 
9.97  580 
9.97  576 
9.97  571 


9.97  567 


L.  Cos.      d.       L.  Cot.     c.d.     L.  Tan.      L.  Sin 


71° 


P.P. 


43 

3.6 

7.2 

I0.8 

14.3 
17.9 


3.4 
6.8 
10.2 


30121.5 


25.1 
28.7 
32.2 
35.8 
39-4 


42 

3.5 

7.0 
10.5 
14.0I13.7 
17.5117.1 
21.0  20.5 

24.5  23.9 
28.0:27.3 
3I.S30.8 
35.0  34.2 
38.5'37.6 


39     38     37 


3.2 
6.5 
9.8 
13.0 
16.2 
19-5 
22.8 
4O126.O 
45  39.2I28.5 
5032.5131.7 
55'35.8'34.8 


3.2 
6.3 
9.5 
12.7 
15.8 
19.0 
22.2 
25.3 


3.1 

6.2 

9.2 

12.3 

15.4 
18.5 
21.6 
24.7 
27.8 
30.8 
33.9 


"  I  36  I  5  I  4 
51   3.00.4,0.3 


10    6.0  0.8 

0.7 

15'    9-0  1.2 

I.O 

20  12.0  1.7 

1.3 

25  15.0  2.1 

1.7 

30  18.0  2.5 

2.0 

35  21.02.9 

2.3 

40  24.0  3.3 

2.7 

45,27.03.8 

3.0 

so'30.0  4.2 

3.3 

5533.04.6 

3.7 

P.P. 


19° 


143 


10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


d. 


,51  264 


51  301 
.51  338 
51374 
51411 
51447 
51  484 
51  520 
51  557 
51  593 


9.51629 


.51  666 
.51  702 
.51  738 
.51  774 
.51811 
,51  847 
,51  883 
51919 
51955 


9.51991 


.52  027 
.52  063 
.52  099 
.52  135 
.52  171 
.52  207 
.52  242 
.52  278 
.52  314  I 


9.52  350 


.52  385 
.52  421 
.52  456 
.52  492 
.52  527 
.52  563 
.52  598 
.52  634 
.52  669  ! 


9.52  705 


.52  740 
.52  775 
.52  811 
.52  846 
.52  881 
.52  916 
.52  951 
.52  986 
.53  021 


9.53  056  I 


53  092 
53  126 
53  161 
53  196 
53  231 
53  266 
53  301 
53  336 
,53  370 


53  405 


L.  Cos. 


37 
37 
36 
37 
36 
37 
36 
37 
36 
36 
37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 


L.  Tan.  c.d.  L.  Cot. 


9.53  697 


9.53  738 
9.53  779 
9.53  820 
9.53  861 
9.53  902 
9.53943 

9.53  984 

9.54  025 
9.54  065 


9.54  106 


9,54  147 
9.54  187 
9.54  228 
9.54  269 
9.54  309 
9.54  350 
9.54  390 
9.54  431 
9.54  471 


9.54  512 


9.54  552 
9.54  593 
9.54  633 
9.54  673 
9.54  714 
9.54  754 
9.54  794 
9.54  835 
9.54  875 


9.54  915 


9.54  955 
9.54  995 
55  035 
55  075 
55  115 
55  155 
55  195 
55  235 
55  275 


9.55  315 


9.55  355 
9.55  395 
9.55  434 
9.55  474 
9.55  514 
9.55  554 
9.55  593 
9.55  633 
9.55  673 


9.55  712 


9.55  752 
9.55  791 
9.55  831 
9.55  870 
9.55  910 
9.55  949 

9.55  989 

9.56  028 
9.56  067 


9.56  107 


41 
41 
41 
41 
41 
41 
41 
41 
40 
41 
41 
40 
41 
41 
40 
41 
40 
41 
40 
41 
40 
41 
40 
40 
41 
40 
40 
41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
39 
40 
40 
40 
39 
40 
40 
39 
40 
39 
40 
39 
40 
39 
40 
39 
39 
40 


L.  Cot.  I  c.d. 


0.46  303 
0.46  262 
0.46  221 
0.46  180 
9.46  139 
0.46  098 
0.46  057 
0.46  016 
0.45  975 
0.45  935 


0.45  894 


0.45  853 
0.45  813 
0.45  772 
0.45  731 
0.45  691 
0.45  650 
0.45  610 
0.45  569 
0.45  529 


0.45  488 


0.45  448 
0.45  407 
0.45  367 
0.45  327 
0.45  286 
0.45  246 
0.45  206 
0.45  165 
0.45  125 
0.45  085 


0.45  045 
0.45  005 
0.44  965 
0.44  925 
0.44  885 
0.44  845 
0.44  805 
0.44  765 
0.44  725 


0.44  685 


0.44  645 
0.44  605 
0.44  566 
0.44  526 
0.44  486 
0.44  446 
0.44  407 
0.44  367 
0.44  327 
0.44  288 


0.44  248 
0.44  209 
0.44  169 
0.44  130 
0.44  090 
0.44  051 
0.44  011 
0.43  972 
0.43  933 


0.43  893 


L.  Cos. 


9.97  567 


9.97  563 
9.97  558 
9.97  554 
9.97  550 
9.97  545 
9.97  541 
9.97  536 
9.97  532 
9.97  528 


9.97  523 


9.97  519 
9.97  515 
9.97  510 
9.97  506 
9.97  501 
9.97  497 
9.97  492 
9.97  488 
9.97  484 


9.97  479 


9.97  475 
9.97  470 
9.97  466 
9.97  461 
9.97  457 
9.97  453 
9.97  448 
9.97  444 
9.97  439 


9.97  435 


9.97  430 
9.97  426 
9.97  421 
9.97  417 
9.97  412 
9.97  408 
9.97  403 
9.97  399 
9.97  394 


9.97  390 


9.97  385 
9.97  381 
9.97  376 
9.97  372 
9.97  367 
9.97  363 
9.97  358 
9.97  353 
9.97  349 


9.97  344 


9.97  340 
9.97  335 
9.97  331 
9.97  326 
9.97  322 
9.97  317 
9.97  312 
9.97  308 
9.97  303 


9.97  299 


L.  Tan.      L.  Sin. 
7(V 


d. 


40 

39 

38 
37 
36 
35 
34 
33 
32 

il 
30 

29 

28 
27 
26 
25 
24 
23 
22 

11 
20 


P.P. 


41  I  40  I    39 


34 
6.8 

I0.2 

13-7 
I7.I 
20.5 
23-9 
27.3 
30.8 
34.2 
37.6 


I  3.3;  3-2 
I  6.7|  6.5 
lO.O  9.8 
,13-3  13-0 
,16.7  16.2 
20.0  19.5 
23.3'22.8 
26.7  j  26.0 
30.0  29.2 
33.3'32.S 
36.735.8 


37     36  ,  35 


3-1 
6.2 
9.2 
12.3 
15.4 
30|i8.5 
3521.6 
40,24.7 
45:27.8 
50I30.8 
5533-9 


3.0,  2.9 
6.0  5-8 
9.0  8.8 
12.0  11.7 
15.01 14.6 
18.0  17.5 
21.0  20.4 
24.0I23.3 
27.0  26.2 
30.0  29.2 
33-ol32.i 


IM 


0.40.3 
0.8:0.7 


34  I  5  I  4 

2.8 

5-7 

8.5 
11.3 
14.2 
17.0 
19.8 
22.7 
25.5 
28.3 


15 
20 
25 
30 
35 
40 
45 
50 
5531.2 


P.P. 


144  20° 


1 

2 
3 
4 
5 
6 
7 
8 
_9 

11 
12 
13 
14 
15 
16 
17 
18 

20 


21 
22 
23 
24 
25 
26 
27 
28 
_29. 
30 


31 
32 
33 
34 
35 
36 
37 
38 

41 
42 
43 
44 
45 
46 
47 
48 
49^ 
50 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos. 


9.53  405 


9.53  440 
9.53  475 
9.53  509 
9.53  544 
9.53  578 
9.53  613 
9.53  647 
9.53  682 
9.53  716 


9.53  751 


9.53  785 
9.53  819 
9.53  854 
9.53  888. 
9.53  922 
9.53  957 

9.53  991 

9.54  025 
9.54  059 


9.54  093 


9.54  127 
9.54  161 
9.54  195 
9.54  229 
9.54  263 
9.54  297 
9.54  331 
9.54  365 
9.54  399 


9.54  433 


9.54  466 
9.54  500 
9.54  534 
9.54  567 
9.54  601 
9.54  635 
9.54  668 
9.54  702 
9.54  735 


9.54  769 


9.54  802 
9.54  836 
9.54  869 
9.54  903 
9.54  936 

9.54  969 

9.55  003 
9.55  036 
9.55  069 


9.55  102 


55  136 
55  169 
55  202 
55  235 
55  268 
55  301 
55  334 
55  367 
55  400 


55  433 


L.  Cos. 


35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 


9.56  107 


9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 
9.56 


146 
185 
224 
264 
303 
342 
381 
420 
459 


9.56  887 


9.56 
9.56 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 


926 
965 
004 
042 
081 
120 
158 
197 
235 


9.57  274 


9.57 
9.57 

57 
57 
57 
57 
57 
57 


9.57 


312 
351 
389 
428 
466 
504 
543 
581 
619 


9.57  658 


9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.57 
9.58 


696 
734 
772 
810 
849 
887 
925 
963 
001 


9.58  039 


9.58 
9.58 
9.58 
9.58 
9.58 
9.58 
9.58 
9.58 
9.58 


077 
115 
153 
191 
229 
267 
304 
342 
380 


9.58  418 


d.   L.  Cot.  c.d. 


39 
39 
39 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
38 
39 
39 
39 
38 
39 
39 
38 
39 
38 
39 
38 
39 
38 
39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
39 
38 
38 
38 
38 
3S 
38 
38 
38 
38 
38 
38 
37 
38 
38 
38 


0.43  893 


0.43  854 
0.43  815 
0.43  776 
0.43  736 
0.43  697 
0.43  658 
0.43  619 
0.43  580 
0.43  541 


0.43  502 


0.43  463 
0.43  424 
0.43  385 
0.43  346 
0.43  307 
0.43  268 
0.43  229 
0.43  190 
0.43  151 


0.43  113 


0.43  074 
0.43  035 
0.42  996 
0.42  958 
0.42  919 
0.42  880 
0.42  842 
0.42  803 
0.42  765 


0.42  726 


0.42  688 
0.42  649 
0.42  611 
0.42  572 
0.42  534 
0.42  496 
0.42  457 
0.42  419 
0.42  381 


0.42  342 


0.42  304 
0.42  266 
0.42  228 
0.42  190 
0.42  151 
0.42  113 
0.42  075 
0.42  037 
0.41  999 


0.41  961 


0.41  923 
0.41  885 
0.41  847 
0.41  809 
0.41  771 
0.41  733 
0.41  696 
0.41  658 
0.41  620 


0.41  582 


9.97  299 


9.97  294 
9.97  289 
9.97  285 
9.97  280 
9.97  276 
9.97  271 
9.97  266 
9.97  262 
9.97  257 


9.97  252 


9.97  248 
9.97  243 
9.97  238 
9.97  234 
9.97  229 
9.97  224 
9.97  220 
9.97  215 
9.97  210 


9.97  206 


9.97  201 
9.97  196 
9.97  192 
9.97  187 
9.97  182 
9.97  178 
9.97  173 
9.97  168 
9.97  163 


9.97  159 


9.97  154 
9.97  149 
9.97  145 
9.97  140 
9.97  135 
9.97  130 
9.97  126 
9.97  121 
9.97  116 


9.97  111 


9.97  107 
9.97  102 
9.97  097 
9.97  092 
9.97  087 
9.97  083 
9.97  078 
9.97  073 
9.97  068 


9.97  063 


9.97  059 
9.97  054 
9.97  049 
9.97  044 
9.97  039 
9.97  035 
9.97  030 
9.97  025 
9.97  020 


9.97  015 


L.  Tan.   L.  Sin. 


d. 


d. 


59 

58 
57 
56 
55 
54 
53 
52 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
31 
3)^ 
30 
29 
28 
27 
26 
25 
24 
23 
22 

IL 

20^ 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


P.P. 


"  I  40  39  38 


5  3-3 
10  6.7 
is'io.o 


13.3 
16.7 
20.0 
,23.3 
'26.7 
30.0 
50133.3 
55 136.7 


3-2  3-2 
6.5  6.3 
9.8    9-5 


13.0 
16.2 
19.5 
22.8 
26.0 
29.2 
32.5 
35.8 


12.7 
IS.8 
19.0 
22.2 
25.3 
28.5 
31.7 
34.8 


37 

35 

3.1 

2.0 

6.2 

5.8 

9.2 

8.8 

12.3 

II.7! 

15.4 

14.6 

18.5117.51 

21.6l20.4l 

24.7 

23.3 

27.8 

26.2 

30.8 

29.2 

33.9 

32.1 

34 

2.8 
5.7 
8.5 
11.3 
14.2 
17.0 
19.8 
22.7 
25-5 
28.3 
31.2 


\  5  I  4 
.8  0.4  0.3 
,5  0.8  0.7 
,2  1.2 
oji.7 
.8  2.1 
5  2.5 
,2  2.9 
03.3 
,83.8 
,54.2 
,2  4.6 


P.P. 


ai\o 


21 


145 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


31 

32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49. 


50 

51 
52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos. 


9.55  433 


9.55  466 
9.55  499 
9.55  532 
9.55  564 
9.55  597 
9.55  630 
9.55  663 
9.55  695 
9.55  728 


9.55  761 


9.55  793 
9.55  826 
9.55  858 
9.55  891 
9.55  923 
9.55  956 

9.55  988 

9.56  021 
9.56  053 


9.56  085 


9.56  118 
9.56  150 
9.56  182 
9.56  215 
9.56  247 
9.56  279 
9.56  311 
9.56  343 
9.56  375 


9.56  408 


9.56  440 
9.56  472 
9.56  504 
9.56  536 
9.56  568 
9.56  599 
9.56  631 
9.56  663 
9.56  695 


9.56  727 


9.56  759 
9.56  790 
9.56  822 
9.56  854 
9.56  886 
9.56  917 
9.56  949 

9.56  980 

9.57  012 


9.57  044 


9.57  075 
9.57  107 
9.57  138 
9.57  169 
9.57  201 
9.57  232 
9.57  264 
9.57  295 
9.57  326 


9.57  358 


L.  Cos. 


33 
33 
33 
32 
33 
33 
33 
31 
33 
33 
31 
33 
31 
33 
32 
33 
31 
33 
31 
31 
33 
31 
31 
33 
31 
31 
31 
31 
31 
33 
31 
31 
31 
31 
31 
31 
32 
32 
32 
32 
32 
31 
32 
32 
32 
31 
32 
31 
32 
32 
31 
32 
31 
31 
32 
31 
32 
31 
31 
32 


58  418 


58  455 
58  493 
58  531 
58  569 
58  606 
58  644 
58  681 
58  719 
58  757 


58  794 


58  832 
58  869 
58  907 
58  944 

58  981 

59  019 
59  056 
59  094 
59  131 


59  168 


59  205 
59  243 
59  280 
59  317 
59  354 
59  391 
59  429 
59  466 
59  503 


59  540 


59  577 
59  614 
59  651 
59  688 
59  725 
59  762 
59  799 
59  835 
59  872 


59  909 


59  946 

59  983 

60  019 
60  056 
60  093 
60  130 
60  166 
60  203 
60  240 


60  276 


60  313 
60  349 
60  386 
60  422 
60  459 
60  495 
60  532 
60  568 
60  605 


60  641 


d.   L.  Cot. 


37 
38 
38 
38 
37 
3^ 
37 
38 
3^ 
37 
38 
37 
38 
37 
37 
3Z 
31 
38 
37 
37 
37 
38 
37 
37 
37 
37 
3'^ 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
36 
37 
37 
37 
37 
36 
37 
37 
37 
36 
37 
37 
36 
37 
36 
37 
36 
37 
36 
37 
36 
37 
36 


41  582 


41  545 
41  507 
41  469 
41  431 
41  394 
41  356 
41  319 
41  281 
41  243 


.41  206 


41  168 
41  131 
41093 
41056 
41019 
40  981 
40  944 
40  906 
40  869 


40  832 


40  795 
40  757 
40  720 
40  683 
40  646 
40  609 
40  571 
40  534 
40  497 


40  460 


40  423 
40  386 
40  349 
40  312 
40  275 
40  238 
40  201 
40  165 
40  128 


40  091 


40  054 
40  017 
39  981 
39  944 
39  907 
39  870 
39  834 
39  797 
39  760 


39  724 


39  687 
39  651 
39  614 
39  578 
39  541 
39  505 
39  468 
39  432 
39  395 


39  359 


9.97  015 


9.97  010 
9.97  005 
9.97  001 
9.96  996 
9.96  991 
9.96  986 
9.96  981 
9.96  976 
9.96  971 


9.96  966 


9.96  962 
9.96  957 
9.96  952 
9.96  947 
9.96  942 
9.96  937 
9.96  932 
9.96  927 
9.96  922 


9.96  917 


9.96  912 
9.96  907 
9.96  903 
9.96  898 
9.96  893 
9.96  888 
9.96  883 
9.96  878 
9.96  873 


9.96  868 


9.96  863 
9.96  858 
9.96  853 
9.96  848 
9.96  843 
9.96  838 
9.96  833 
9.96  828 
9.96  823 


9.96  818 


9.96  813 
9.96  808 
9.96  803 
9.96  798 
9.96  793 
9.96  788 
9.96  783 
9.96  778 
9.96  772 


9.96  767 


9.96  762 
9.96  757 
9.96  752 
9.96  747 
9.96  742 
9.96  737 
9.96  732 
9.96  727 
9.96  722 


9.96  717 


10 


c.d.    L.  Tan.      L.  Sin 

68° 


d. 


60 

59 

58 
57 
56 

55 
54 
53 
52 

50 

49 
48 
47 
46 
45 
44 
43 
42 

IL 
40 

39 

38 
37 
36 
35 
34 
33 
31 

29 

28 
27 
26 
25 
24 
23 
22 

11. 
20 


P.P. 


38  I  37     36 


St  3-2] 
10^  6.3 

IS   p-s! 

20  12.7 

25  15.81 

30  IQ.O 
35  22.2 


3-0 
6.0 
9.0 


25-3 
28.5 
31.7 
34-8 


12.3   12.0 

15-4  15.0 

18.5  18.0 

21.6  21.0 

24.7  24.0 

27.8  27.0 
30.8|30.o 
33.9i33.0 


33     32     31 


2.8 
5.5 
8.2 
II.O 

13.8 
16.5 

19.2 

22.0 
24.8 
27.5 
30.2 


2.7 
5.3 

8.0 
10.7 
13.3 
16.0 
18.7 
21.3 
24.0 
26.7 


2.6 

5-2 

7.8 
10.3 

12.9 

I5.S 
I8.I 
20.7 
23.2 

25.8 


29.3128.4 


"I  6 

SO.s 
10  I. o 
15,1.5 
20  2.0 
25  2.5 
3030 
35  3-5 
40 '4.0 
45*4.5 
50,5.0 
S5iS.5 


5     4 


P.P. 


20 

21 
22 
23 
24 
25 
26 
27 
28 

30 


40 

41 
42 
43 
44 
45 
46 
47 
48 

50 

51 

52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d 


9.57  358 


9.57  389 
9.57  420 
9.57  451 
9.57  482 
9.57  514 
9.57  545 
9.57  576 
9.57  607 
9.57  638 


9.57  669 


9.57  700 
9.57  731 
9.57  762 
9.57  793 
9.57  824 
9.57  855 
9.57  885 
9.57  916 
9.57  947 


9.57  978 


9.58  008 
9.58  039 
9.58  070 
9.58  101 
9.58  131 
9.58  162 
9.58  192 
9.58  223 
9.58  253 


9.58  284 


9.58  314 
9.58  345 
9.58  375 
9.58  406 
9.58  436 
9.58  467 
9.58  497 
9.58  527 
9.58  557 


9.58  588 


9.58  618 
9.58  648 
9.58  678 
9.58  709 
9.58  739 
9.58  769 
9.58  799 
9.58  829 
9.58  859 


9.58  889 


58  919 
58  949 

58  979 

59  009 
59  039 
59  069 
59  098 
59  128 
59  158 


59  188 


31 
31 
31 
31 
32 
31 
31 
31 
31 
31 
31 
31 

a 

31 
31 
31 
30 
31 
31 
31 
30 
31 
31 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 
30 
30 
31 
30 
30 
30 
31 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
30 


L.  Cos. 


9.60  641 


9.60 
9.60 
9.60 
9.60 
9.60 
9.60 
9.60 
9.60 
9.60 


677 
714 
750 
786 
823 
859 
895 
931 
967 


9.61  004 


9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 


040 
076 
112 
148 
184 
220 
256 
292 
328 


9.61  364 


9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 


400 
436 
472 
508 
544 
579 
615 
651 
687 


9.61  722 


9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.61 
9.62 
9.62 


758 
794 
830 
865 
901 
936 
972 
008 
043 


9.62  079 


9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 


114 
150 
185 
221 
256 
292 
327 
362 
398 


9.62  433 


9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 
9.62 


9.62 


468 
504 
539 
574 
609 
645 
680 
715 
750 
'785 


d.   L.  Cot.  c.d 


36 
37 
36 
36 
37 
36 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 
35 
36 
36 
36 
35 
36 
35 
36 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
36 
35 
35 
35 
36 
35 
35 
35 
35 


0.39  359 


0.39  323 
0.39  286 
0.39  250 
0.39  214 
0.39  177 
0.39  141 
0.39  105 
0.39  069 
0.39  033 


0.38  996 


0.38  960 
0.38  924 
0.38  888 
0.38  852 
0.38  816 
0.38  780 
0.38  744 
0.38  708 
0.38  672 


0.38  636 


0.38  600 
0.38  564 
0.38  528 
0.38  492 
0.38  456 
0.38  421 
0.38  385 
0.38  349 
0.38  313 


0.38  278 


0.38  242 
0.38  206 
0.38  170 
0.38  135 
0.38  099 
0.38  064 
0.38  028 
0.37  992 
0.37  957 


0.37  921 


0.37  886 
0.37  850 
0.37  815 
0.37  779 
0.37  744 
0.37  708 
0.37  673 
0.37  638 
0.37  602 
0.37  567 


0.37  532 
0.37  496 
0.37  461 


0.37  426 
0.37  391 
0.37  355 
0.37  320 
0.37  285 
0.37  250 


0.37  215 


9.96  717 


9.96  711 
9.96  706 
9.96  701 
9.96  696 
9.96  691 
9.96  686 
9.96  681 
9.96  676 
9.96  670 


9.96  665 


9.96  660 
9.96  655 
9.96  650 
9.96  645 
9.96  640 
9.96  634 
9.96  629 
9.96  624 
9.96  619 


9.96  614 


9.96  608 
9.96  603 
9.96  598 
9.96  593 
9.96  588 
9.96  582 
9.96  577 
9.96  572 
9.96  567 


9.96  562 


9.96  556 
9.96  551 
9.96  546 
9.96  541 
9.96  535 
9.96  530 
9.96  525 
9.96  520 
9.96  514 


9.96  509 


9.96  504 
9.96  498 
9.96  493 
9.96  488 
9.96  483 
9.96  477 
9.96  472 
9.96  467 
9.96  461 


9.96  456 


9.96  451 
9.96  445 
9.96  440 
9.96  435 
9.96  429 
9.96  424 
9.96  419 
9.96  413 
9.96  408 


9.96  403 


L.  Tan.   L.  Sin. 


67' 


d. 


60 


59 

58 
57 
56 
55 
54 
53 
52 

ii 
50 

49 

48 
47 
46 
45 
44 
43 
42 

40^ 

39 

38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 

IL 
^0 

19 
18 
17 
16 
15 
14 
13 
12 
J_l 

9 

8 
7 
6 
5 
4 
3 
2 
1 
0 


P.P. 


"  I  37  !  36     35 


S    3.li 
10     6.2i 

IS'  9.2: 
20  12.3 
25  1S.4 
30  18. 5 
35  21.6 
40,24.71 
45  27.8| 
SO  30.8 
55  33.9I 


2.9 
5.8 
8.8 


12.0  11.7 
15.0  14.6 
18.0  17.5 
21.0  20.4 
24.0  23.3 
27.0  26.2 
30.0  29.2 
33.032.1 


"I  32 
5  2.7 
10!  5.3 
15I  8.0 
20!  10.7  j 
25  13-3 
3oii6.o 
35  18.71 
4021.3! 
45  24.0 
SO  26.71 
55  29.3I 


31      30 


2.6 
5-2 

7.8 
10.3 

12.9 
15-5 
18. 1 


2.5 
5.0 
7.5 
10.0 
12.S 
150 
17-5 


20.7120.0 

23.2'22.5 
25.8  25.0 
28.4I27.S 


II 

29 

6j5 

5 

2.4 

0.5  0.4 

10 

4.8 

i.o  0.8 

IS 

7.2 

i.5il.2 

20 

9.7 

2.0,1.7 

25 

12. 1 

2.52.1 

30 

14. 5 

2.0  2.5 

35 

16.9 

3.5  2.9 

40  19.3 

403.3 

45 

21.8 

4-5  3.8 

SO  24.2 

5-0  4-2 

55 

26.6 

5.5  4.6 

P.P. 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


41 
42 
43 
44 
45 
46 
47 
48 
49 

51 
52 
53 
54 
55 
56 
57 
58 
59 


L.  Sin. 


9.59  188 


9.59  218 
9.59  247 
9.59  277 
9.59  307 
9.59  336 
9.59  366 


9.59  484 


9.59  514 
9.59  543 
9.59  573 
9.59  602 
9.59  632 
9.59  661 
9.59  690 
9.59  720 
9.59  749 


d.   L.  Tan.  c.d. 


9.59  778 


9.59  808 
9.59  837 
9.59  866 
9.59  895 
9.59  924 
9.59  954 

9.59  983 

9.60  012 
9.60  041 


9.60  070 


9.60  099 
9.60  128 
9.60  157 
9.60  186 
9.60  215 
9.60  244 
9.60  273 
9.60  302 
9.60  331 


9.60  359 


9.60  388 
9.60  417 
9.60  446 
9.60  474 
9.60  503 
9.60  532 
9.60  561 
9.60  589 
9.60  618 


9.60  646 


9.60  675 
9.60  704 
9.60  732 
9.60  761 
9.60  789 
9.60  818 
9.60  846 
9.60  875 
9.60  903 


9.60  931 


30 

29 

30 

30 

29 

30 

30 

29 

30 

29 

30 

29 

30 

29 

30 

29 

29 

30 

29 

29 

30 

29 

29 

29 

29 

30 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

28 

29 

29 

29 

28 

29 

29 

29 

28 

29 

28 

29 

29 

28 

29 

28 

29 

28 

29 

28 

28 


9.62  785 


9.62  820 
9.62  855 
9.62  890 
9.62  926 
9.62  961 

9.62  996 

9.63  031 
9.63  066 
9.63  101 


9.63  135 


9.63  170 
9.63  205 
9.63  240 
9.63  275 
9.63  310 
9.63  345 
9.63  379 
9.63  414 
9.63  449 


9.63  484 


9.63  519 
9.63  553 
9.63  588 
9.63  623 
9.63  657 
9.63  692 
9.63  726 
9.63  761 
9.63  796 


9.63  830 


9.63  865 
9.63  899 
9.63  934 

9.63  968 

9.64  003 
9.64  037 
9.64  072 
9.64  106 
9.64  140 


9.64  175 


9.64  209 
9.64  243 
9.64  278 
9.64  312 
9.64  346 
9.64  381 
9.64  415 
9.64  449 
9.64  483 


9.64  517 


9.64  552 
9.64  586 
9.64  620 
9.64  654 
9.64  688 
9.64  722 
9.64  756 
9.64  790 
9.64  824 


9.64  858 


35 
35 
35 
36 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 


23° 


L.  Cot. 


147 


0.37  215 


0.37  180 
0.37  145 
0.37  110 
0.37  074 
0.37  039 
0.37  004 
0.36  969 
0.36  934 
0.36  899 


0.36  865 


0.36  830 
0.36  795 
0.36  760 
0.36  725 
0.36  690 
0.36  655 
0.36  621 
0.36  586 
0.36  551 


0.36  516 


0.36  481 
0.36  447 
0.36  412 
0.36  377 
0.36  343 
0.36  308 
0.36  274 
0.36  239 
0.36  204 


0.36  170 


0.36  135 
0.36  101 
0.36  066 
0.36  032 
0.35  997 
0.35  963 
0.35  928 
0.35  894 
0.35  860 


0.35  825 


0.35  791 
0.35  757 
0.35  722 
0.35  688 
0.35  654 
0.35  619 
0.35  585 
0.35  551 
0.35  517 


0.35  483 


0.35  448 
0.35  414 
0.35  380 
0.35  346 
0.35  312 
0.35  278 
0.35  244 
0.35  210 
0.35  176 


0.35  142 


L.  Cos. 


d. 


9.96  403 


9.96  397 
9.96  392 
9.96  387 
9.96  381 
9.96  376 
9.96  370 
9.96  365 
9.96  360 
9.96  354 


9.96  349 


9.96  343 
9.96  338 
9.96  333 
9.96  327 
9.96  322 
9.96  316 
9.96  311 
9.96  305 
9.96  300 


9.96  294 


9.96  289 
9.96  284 
9.96  278 
9.96  273 
9.96  267 
9.96  262 
9.96  256 
9.96  251 
9.96  245 


9.96  240 


9.96  234 
9.96  229 
9.96  223 
9.96  218 
9.96  212 
9.96  207 
9.96  201 
9.96  196 
9.96  190 


9.96  185 


9.96  179 
9.96  174 
9.96  168 
9.96  162 
9.96  157 
9.96  151 
9.96  146 
9.96  140 
9.96  135 


9.96  129 


9.96  123 
9.96  118 
9.96  112 
9.96  107 
9.96  101 
9.96  095 
9.96  090 
9.96  084 
9.960^79 
9^96  073 


L.  Cos.   d.   L.  Cot.  c.d.  L.  Tan.  L.  Sin.   d. 


6G^ 


P.P. 


" 


36  35  34 


51  3-0 
lOj  6.0 
15  9.0 
20|l2.0  II.7 

25ii5.o  14-6 
30  i8.o  17.5 
3S'2i.o  20.4 
40' 


2.8 

5-7 

8.5 

III.3 

14.2 

17.0 

19.8 


24.0  23.3  22.7 
45  27.0  26.2  25.5 
50  30.0  29.2  28.3 
5533.032.1  31.2 


30  !  29     28 


5    2.5    2.4 
10    5.0i  4.8 

15    7.5    7-2 


20JIO.0 

25|I2.5 

30,15.0 

35  17.5 


20.0 
22.5 
25.0 
27.5 


9.7 
12. 1 
14-5 
16.9 


2.3 
4.7 
7.0 
9.3 
11.7 
14.0 
16.3 


19.3  18.7 
21.8  21.0 
24.2  23.3 
26.6I25.7 


6  I  5 


0.5:0.4 
i.0|0.8 

I.5|1.2 
2.0|1.7 

2.5  2.1 
30  2.5 
3-5  2.9 
40,4  03.3 
45.4.53.8 
50  5.04.2 

ss's.s  4.6 


P.P. 


148 


24° 


10 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 

52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d 


9.60  931 


.60  960 
,60  988 
.61  016 
.61  045 
,61  073 
.61  101 
.61  129 
61  158 
•.61  186 


9.61  214 


.61  242 
.61  270 
.61  298 
,61  326 
,61  354 
.61  382 
.61411 
61  438 
.61  466 


9.61  494 


,61  522 
,61  550 
,61  578 
,61  606 
,61  634 
,61  662 
,61  689 
,61  717 
,61  745 


9.61  773 


.61  800 
.61  828 
.61  856 
.61  883 
.61911 
.61  939 
.61  966 
.61  994 
.62  021 


9.62  049 


.62  076 
.62  104 
.62  131 
.62  159 
.62  186 
.62  214 
.62  241 
.62  268 
,62  296 


,62  323 


,62  350 
,62  377 
.62  405 
,62  432 
,62  459 
,62  486 
,62  513 
,62  541 
,62  568 


9.62  595 


29 

28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
29 
27 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
27 
28 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 
27 
28 
27 
27 


9.64  858 


9.64  892 
9.64  926 
9.64  960 

9.64  994 

9.65  028 
9.65  062 
9.65  096 
9.65  130 
9.65  164 


9.65  197 


9.65  231 
9.65  265 
9.65  299 
9.65  333 
9.65  366 
9.65  400 
9.65  434 
9.65  467 
9.65  501 


9.65  535 


9.65  568 
9.65  602 
9.65  636 
9.65  669 
9.65  703 
9.65  736 
9.65  770 
9.65  803 
9.65  837 


9.65  870 


9.65  904 
9.65  937 

9.65  971 

9.66  004 
9.66  038 
9.66  071 
9.66  104 
9.66  138 
9.66  171 


9.66  204 


9.66  238 
9.66  271 
9.66  304 
9.66  337 
9.66  371 
9.66  404 
9.66  437 
9.66  470 
9.66  503 


9.66  537 


9.66  570 
9.66  603 
9.66  636 
9.66  669 
9.66  702 
9.66  735 
9.66  768 
9.66  801 
9.66  834 


9.66  867 


34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 


0.35  142 


0.35  108 
0.35  074 
0.35  040 
0.35  006 
0.34  972 
0.34  938 
0.34  904 
0.34  870 
0.34  836 


0.34  803 


0.34  769 
0.34  735 
0.34  701 
0.34  667 
0.34  634 
0.34  600 
0.34  566 
0.34  533 
0.34  499 


0.34  465 


0.34  432 
0.34  398 
0.34  364 
0.34  331 
0.34  297 
0.34  264 
0.34  230 
0.34  197 
0.34  163 


0.34  130 


0.34  096 
0.34  063 
0.34  029 
0.33  996 
0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 


0.33  796 


0.33  762 
0.33  729 
0.33  696 
0.33  663 
0.33  629 
0.33  596 
0.33  563 
0.33  530 
0.33  497 


0.33  463 


0.33  430 
0.33  397 
0.33  364 
0.33  331 
0.33  298 
0.33  265 
0.33  232 
0.33  199 
0.33  166 


0.33  133 


9.96  073 


9.96  067 
9.96  062 
9.96  056 
9.96  050 
9.96  045 
9.96  039 
9.96  034 
9.96  028 
9.96  022 


9.96  017 


9.96  011 
9.96  005 
9.96  000 
9.95  994 
9.95  988 
9.95  982 
9.95  977 
9.95  971 
9.95  965 


9.95  960 


9.95  954 
9.95  948 
9.95  942 
9.95  937 
9.95  931 
9.95  925 
9.95  920 
9.95  914 
9.95  908 


9.95  902 


9.95  897 
9.95  891 
9.95  885 
9.95  879 
9.95  873 
9.95  868 
9.95  862 
9.95  856 
9.95  850 


9.95  844 


9.95  839 
9.95  833 
9.95  827 
9.95  821 
9.95  815 
9.95  810 
9.95  804 
9.95  798 
9.95  792 


9.95  786 


9.95  780 
9.95  775 
9.95  769 
9.95  763 
9.95  757 
9.95  751 
9.95  745 
9.95  739 
9.95  733 


9.95  728 


P.P. 


"34  33 

5  2.8  2.8 

10  5-7  5-5 

IS  8.5  8.2 

20  1 1.3  II.O 

25  14-2  13.8 
30  17.0  16.S 
35  19-8  19.2 
40  22.7  22.0 
45  25.5  24.8 
SO  28.3  27.S 
55  31.2  30.2 


5 
10 
15 
20 
25 
30 
35 
40 
45  21.8 
SO  24.2 
55  26.6 


29 

2.4 
4.8 
7.2 
9-71 
12. 1 
14-5 
16.9 
19.3 


28 


27 


2.3 

4.7 

7.0 

9.3'  9.0 
11.7I11.2 
14-0  13.5 
16.3  15-8 
18.7  18.0 
21.0  20.2 

23.3  22.5 

25.7  24.8 


5 

0.5  0.4 

10 

1.00.8 

15 

1.5  1.2 

20 

2.0  1.7 

25 

2.5  2.1 

30 

3.0:2.5 

35 

3-5  2.9 

40 

4-03.3 

45 

4.5i3.8 

50 

5.0  4-2 

55 

5.5  4-6 

L.  Cos. 


L.  Cot.  c.d. 


L.  Tan.   L.  Sin. 

65^ 


P.P. 


25° 


149 


10 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


60 


L.  Sin. 


9.62  595 


9.62  622 
9.62  649 
9.62  676 
9.62  703 
9.62  730 
9.62  757 
9.62  784 
9.62  811 
9.62  838 


9.62  865 


9.62  892 
9.62  918 
9.62  945 
9.62  972 

9.62  999 

9.63  026 
9.63  052 
9.63  079 
9.63  106 


9.63  133 


9.63  159 
9.63  186 
9.63  213 
9.63  239 
9.63  266 
9.63  292 
9.63  319 
9.63  345 
9.63  372 


9.63  398 


9.63  425 
9.63  451 
9.63  478 
9.63  504 
9.63  531 
9.63  557 
9.63  583 
9.63  610 
9.63  636 


9.63  662 


9.63  689 
9.63  715 
9.63  741 
9.63  767 
9.63  794 
9.63  820 
9.63  846 
9.63  872 
9.63  898 


9.63  924 


9.63  950 

9.63  976 

9.64  002 
9.64  028 
9.64  054 
9.64  080 
9.64  106 
9.64  132 
9.64  158 


9.64  184 


d. 


27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 


L.  Tan. 


9.66  867 


9.66  900 
9.66  933 
9.66  966 

9.66  999 

9.67  032 
9.67  065 
9.67  098 
9.67  131 
9.67  163 


9.67  196 


9.67  229 
9.67  262 
9.67  295 
9.67  327 
9.67  360 
9.67  393 
9.67  426 
9.67  458 
9.67  491 


9.67  524 


9.67  556 
9.67  589 
9.67  622 
9.67  654 
9.67  687 
9.67  719 
9.67  752 
9.67  785 
9.67  817 


9.67  850 


9.67  882 
9.67  915 
9.67  947 

9.67  980 

9.68  012 
9.68  044 
9.68  077 
9.68  109 
9.68  142 


9.68  174 


9.68  206 
9.68  239 
9.68  271 
9.68  303 
9.68  336 
9.68  368 
9.68  400 
9.68  432 
9.68  465 


9.68  497 


I  L.  Cos.  I  d.   L.  Cot.  c.d 


9.68  529 
9.68  561 
9.68  593 
9.68  626 
9.68  658 
9.68  690 
9.68  722 
9.68  754 
9.68  786 


9.68  818 


c.d.  L.  Cot. 


33 
33 
33 
33 
33 
33 
33 
33 
31 
33 
33 
33 
33 
31 
33 
33 
33 
31 
33 
33 
31 
33 
33 
31 
33 
31 
33 
33 
31 
33 
31 
33 
31 
33 
31 
31 
33 
31 
33 
31 
31 
33 
31 
31 
33 
31 
31 
31 
33 
31 
31 
31 
31 
33 
31 
31 
31 
32 
32 
32 


0.33  133 


0.33  100 
0.33  067 
0.33  034 
0.33  001 
0.32  968 
0.32  935 
0.32  902 
0.32  869 
0.32  837 


0.32  804 


0.32  771 
0.32  738 
0.32  705 
0.32  673 
0.32  640 
0.32  607 
0.32  574 
0.32  542 
0.32  509 


0.32  476 


0.32  444 
0.32  411 
0.32  378 
0.32  346 
0.32  313 
0.32  281 
0.32  248 
0.32  215 
0.32  183 


0.32  150 


0.32  118 
0.32  085 
0.32  053 
0.32  020 
0.31  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 


0.31  826 


0.31  794 
0.31  761 
0.31  729 
0.31  697 
0.31  664 
0.31  632 
0.31  600 
0.31  568 
0.31  535 


0.31  503 


0.31471 
0.31  439 
0.31  407 
0.31  374 
0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 


0.31  182 


L.  Cos. 


9.95  728 


9.95  722 
9.95  716 
9.95  710 
9.95  704 
9.95  698 
9.95  692 
9.95  686 
9.95  680 
9.95  674 


9.95  668 


9.95  663 
9.95  657 
9.95  651 
9.95  645 
9.95  639 
9.95  633 
9.95  627 
9.95  621 
9.95  615 


9.95  609 


9.95  603 
9.95  597 
9.95  591 
9.95  585 
9.95  579 
9.95  573 
9.95  567 
9.95  561 
9.95  555 


9.95  549 


9.95  543 
9.95  537 
9.95  531 
9.95  525 
9.95  519 
9.95  513 
9.95  507 
9.95  500 
9.95  494 
9.95  488 


9.95  482 
9.95  476 
9.95  470 
9.95  464 
9.95  458 
9.95  452 
9.95  446 
9.95  440 
9.95  434 


9.95  427 


9.95  421 
9.95  415 
9.95  409 
9.95  403 
9.95  397 
9.95  391 
9.95  384 
9.95  378 
9.95  372 


9.95  366 


L.  Tan.   L.  Sin. 


d. 


60 

59 

58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 


P.P. 


33  I  32 

2.8'    2.7 

S-S\  5-3 
8.2I  8.0 
ii.o  10.7 
13-8  13-3 
16.S  16.0 
35!i9.2  18.7 
40  22.0  21.3 
45124-8  24.0 
50  27.5  26.7 
5S'30.2  29.3 


"  I 


27 


2.2 
4-3 
6.5 
8.7 


5,  2.2 
10!  4-5 
15  6.8 
20|  9.0| 
25  II.2  10.8 
30  13.5  13-0 
35  15-8  15-2 
40  18.0  17.3 
45j20.2  19.S 
50  22.5  21.7 
S5I24-8  23.8 


"I  7  I 
s'0.60 


IO;I. 

IS;I. 


20:2.3  2 
25  2.9  2 
30|3S  3 
35  4-il3 
404.7  4 
45  5.2;4 
50  5.8,5 
55  6.4IS 


6  I  5 

50.4 
o  0.8 
51.2 
0  1.7 
52.1 
o  2.5 
5  2.9 
03-3 
53.8 
04.2 
5 '4.6 


P.P. 


150 


1 

2 

3 

4 

5 

6 

7 

8 

_9 

JO 

11 

12 

13 

14 

15 

16 

17 

18 

21 

22 

23 
24 
25 
26 

27 
28 

30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


9.64  184 


9.64  210 
9.64  236 
9.64  262 
9.64  288 
9.64  313 
9.64  339 
9.64  365 
9.64  391 
9.64  417 


9.64  442 


9.64  468 
9.64  494 
9.64  519 
9.64  545 
9.64  571 
9.64  596 
9.64  622 
9.64  647 
9.64  673 


9.64  698 


9.64  724 
9.64  749 
9.64  775 
9.64  800 
9.64  826 
9.64  851 
9.64  877 
9.64  902 
9.64  927 


9.64  953 


9.64  978 

9.65  003 
9.65  029 
9.65  054 
9.65  079 
9.65  104 
9.65  130 
9.65  155 
9.65  180 


9.65  205 


9.65  230 
9.65  255 
9.65  281 
9.65  306 
9.65  331 
9.65  356 
9.65  381 
9.65  406 
9.65  431 


9.65  456 


9.65  481 
9.65  506 
9.65  531 
9.65  556 
9.65  580 
9.65  605 
9.65  630 
9.65  655 
9.65  680 
9.65  705 


26 
26 
26 
26 
25 
26 
26 
26 
26 
25 
26 
26 
25 
26 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
25 
25 
26 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
24 
25 
25 
25 
25 
25 


26^ 

L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


9.68  818 


9.68  850 
9.68  882 
9.68  914 
9.68  946 

9.68  978 

9.69  010 
9.69  042 
9.69  074 
9.69  106 


9.69  138 


9.69  170 
9.69  202 
9.69  234 
9.69  266 
9.69  298 
9.69  329 
9.69  361 
9.69  393 
9.69  425 


9.69  457 


9.69  488 
9.69  520 
9.69  552 
9.69  584 
9.69  615 
9.69  647 
9.69  679 
9.69  710 
9.69  742 


9.69  774 


9.69  805 
9.69  837 
9.69  868 
9.69  900 
9.69  932 
9.69  963 

9.69  995 

9.70  026 
9.70  058 


9.70  089 


9.70  121 
9.70  152 
9.70  184 
9.70  215 
9.70  247 
9.70  278 
9.70  309 
9.70  341 
9.70  372 


9.70  404 


9.70  435 
9.70  466 
9.70  498 
9.70  529 
9.70  560 
9.70  592 
9.70  623 
9.70  654 
9.70  685 


9.70  717 


32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
31 
32 
32 
32 
32 
31 
32 
32 
32 
31 
32 
32 
31 
32 
32 
31 
32 
31 
32 
32 
31 
32 
31 
32 
31 
32 
31 
32 
31 
32 
31 
31 
32 
31 
32 
31 
31 
32 
31 
31 
32 
31 
31 
31 
32 


0.31  182 


0.31  150 
0.31  118 
0.31  086 
0.31  054 
0.31  022 
0.30  990 
0.30  958 
0.30  926 
0.30  894 


0.30  862 
0.30  830 
0.30  798 
0.30  766 
0.30  734 
0.30  702 
0.30  671 
0.30  639 
0.30  607 
0.30  575 


9.95  366 


9.95  360 
9.95  354 
9.95  348 
9.95  341 
9.95  335 
9.95  329 
9.95  323 
9.95  317 
9.95  310 


9.95  304 


0.30  543 


0.30  512 
0.30  480 
0.30  448 
0.30  416 
0.30  385 
0.30  353 
0.30  321 
0.30  290 
0.30  258 


0.30  226 


0.30  195 
0.30  163 
0.30  132 
0.30  100 
0.30  068 
0.30  037 
0.30  005 
0.29  974 
0.29  942 


0.29  911 


0.29  879 
0.29  848 
0.29  816 
0.29  785 
0.29  753 
0.29  722 
0.29  691 
0.29  659 
0.29  628 


0.29  596 


0.29  565 
0.29  534 
0.29  502 
0.29  471 
0.29  440 
0.29  408 
0.29  377 
0.29  346 
0.29  315 


0.29  283 


9.95  298 
9.95  292 
9.95  286 
9.95  279 
9.95  273 
9.95  267 
9.95  261 
9.95  254 
9.95  248 


9.95  242 


9.95  236 
9.95  229 
9.95  223 
9.95  217 
9.95  211 
9.95  204 
9.95  198 
9.95  192 
9.95  185 


9.95  179 


9.95  173 
9.95  167 
9.95  160 
9.95  154 
9.95  148 
9.95  141 
9.95  135 
9.95  129 
9.95  122 


9.95  116 


9.95  110 
9.95  103 
9.95  097 
9.95  090 
9.95  084 
9.95  078 
9.95  071 
9.95  065 
9.95  059 


9.95  052 


9.95  046 
9.95  039 
9.95  033 
9.95  027 
9.95  020 
9.95  014 
9.95  007 
9.95  001 
9.94  995 


9.94  988 


P.P. 


"  I  32  I  31 

s!  2.7!  2.6 

10  5-3'  S.2 

15  8o  7.8 


2010.7 

10.3 

25II3.3 

12.9 

30  16.0 

15.5 

35'i8.7 

18.1 

40  21.3 

20.7 

45  24.0 

23.2 

SO  26.7 

25.8 

55  29.3 

28.4 

"1  26 

25 

24 

S     2.2I    2.1 

2.0 

10    4.31    4-2 
IS    6.5    6.2 
20    8.71   8.3 

4.0 
6.0 
8.0 

25  10.8  10.4,10.0 

30  13.0 
3515.2 
40  17.3 
45|I9.S 
50121.7 

12.5112.0 
14.6  14.0 
16.7I16.0 
i8.8|i8.0 

20.8I20.0 

SS'23.8 

22.9 

22.0 

"I  7 

50.6 

10:1.2 

is;i.8 
20 


2.3I2.0 
2.92.S 

353.0 

4.ii3.5 
4.74.0 
5.24.5 
5.85.0 
6.4I5.5 


L.  Cos. 


d. 


L.  Cot.  c.d.  I  L.  Tan.  |  L.  Sin. 


P.P. 


27° 


151 


1 ' 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

1 

2 
3 

9.65  705 

24 
25 
25 
25 
24 
25 
25 
24 
25 
25 
24 
25 
24 
25 
25 
24 
25 
24 
25 
24 
24 
25 
24 
25 
24 
24 
25 
24 
24 
25 
24 
24 
24 
24 
25 
24 
24 
24 
24 
24 
24 
25 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 

9.70  717 

31 
31 
31 
31 
32 
31 
31 
31 
31 
31 
31 
31 
31 
32 
31 
31 
31 
31 
31 

31 
31 
31 
30 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
31 

3,0 
31 
31 
30 
31 
31 
31 
30 
31 
31 
30 
31 
30 
31 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 

0.29  283 

9.94  988 

6 

7 
6 
7 
6 
7 
6 
7 
6 
7 
6 
6 
7 
6 
7 
6 
7 
7 
6 
7 
6 
7 
6 
7 
6 
7 
6 
7 
7 
6 
7 
6 
7 
6 
7 
7 
6 
7 
6 
7 
7 
6 
7 
7 
6 
7 
7 
6 
7 
7 
6 
7 
7 
6 
7 
7 
6 
7 
7 
7 

60 

59 

58 
57 

9.65  729 
9.65  754 
9.65  779 

9.70  748 
9.70  779 
9.70  810 

0.29  252 
0.29  221 
0.29  190 

9.94  982 
9.94  975 
9.94  969 

4 
5 
6 

9.65  804 
9.65  828 
9.65  853 

9.70  841 
9.70  873 
9.70  904 

0.29  159 
0.29  127 
0.29  096 

9.94  962 
9.94  956 
9.94  949 

56 

55 
54 

"  i  32  1  31  1  30 

7 
8 
9 

9.65  878 
9.65  902 
9.65  927 

9.70  935 
9.70  966 
9.70  997 

0.29  065 
0.29  034 
0.29  003 

9.94  943 
9.94  936 
9.94  930 

53 
52 
51 
50 

49 

48 
47 

s!  2.7  2.6 
10;  S.3|  5-2 
15  8.01  7-8 

20|  10.7  10.3 

25,13.3  12.9 
3016.015.5 
35  i8.7!i8.i 
40  21.3  20.7 
45  24.0  23.2 
50  26.7  25.8 

-5 
5.0 
7.5 

lO.O 

12.5 
15.0 
17.S 
20.0 

22.5 
9t;  n 

10 

11 
12 
13 

9.65  952 

9.71  028 

0.28  972 

9.94  923 

9.65  976 

9.66  001 
9.66  025 

9.71  059 
9.71  090 
9.71  121 

0.28  941 
0.28  910 
0.28  879 

9.94  917 
9.94  911 
9.94  904 

14 
15 
16 

9.66  050 
9.66  075 
9.66  099 

9.71  153 
9.71  184 
9.71215 

0.28  847 
0.28  816 
0.28  785 

9.94  898 
9.94  891 
9.94  885 

46 
45 
44 

55  29.3  28.4  27.S 

17 
18 
19 
20 

21 

22 
23 

9.66  124 
9.66  148 
9.66  173 

9.71  246 
9.71277 
9.71  308 

0.28  754 
0.28  723 
0.28  692 

9.94  878 
9.94  871 
9.94  865 

43 
42 
41 
40 

39 

38 
37 

9.66  197 

9.71339 

0.28  661 

9.94  858 

9.66  221 
9.66  246 
9.66  270 

9.71  370 
9.71  401 
9.71431 

0.28  630 
0.28  599 
0.28  569 

9.94  852 
9.94  845 
9.94  839 

24 
25 
26 

9.66  295 
9.66  319 
9.66  343 

9.71  462 
9.71  493 
9.71  524 

0.28  538 
0.28  507 
0.28  476 

9.94  832 
9.94  826 
9.94  819 

36 
35 
34 

"  1  25  1  24 

23 

27 
28 
29 

9.66  368 
9.66  392 
9.66  416 

9.71  555 
9.71  586 
9.71617 

0.28  445 
0.28  414 
0.28  383 

9,94  813 
9.94  806 
9.94  799 

33 
32 
31 
30 

29 

28 
27 

s 

10 

IS 
20 

2.1 
4.2 
6.2 
8.3 

2.0 
4.0 

6.0 
8.0 

1.9 

3f 
5.8 

7.7 
9.6 

TT  C 

30 

9.66  441 

9.71  648 

0.28  352 

9.94  793 

2S|  10.41  lO.O 
30  12.5  12.0 

31 

32 
33 

9.66  465 
9.66  489 
9.66  513 

9.71679 
9.71  709 
9.71  740 

0.28  321 
0.28  291 
0.28  260 

9.94  786 
9.94  780 
9.94  773 

35  14-6 
40  16.7 
4518.8 
50  20.8 

14.0  13.4 
16.0  15.3 
18.017.2 
20.0  10.2 

34 
35 
36 

9.66  537 
9.66  562 
9.66  586 

9.71  771 
9.71  802 
9.71  833 

0.28  229 
0.28  198 
0.28  167 

9.94  767 
9.94  760 
9.94  753 

26 

25 
24 

55  22.9'22.0;2I.I 

37 
3S 
39 

9.66  610 
9.66  634 
9.66  658 

9.71  863 
9.71  894 
9.71  925 

0.28  137 
0.28  106 
0.28  075 

9.94  747 
9.94  740 
9.94  734 

23 
22 
21 
20 
19 
18 
17 

40 

9.66  682 

9.71  955 

0.28  045 

9.94  727 

41 
42 
43 

9.66  706 
9.66  731 
9.66  755 

9.71986 
9.72  017 
9.72  048 

0.28  014 
0.27  983 
0.27  952 

9.94  720 
9.94  714 
9.94  707 

44 
45 
46 

9.66  779 
9.66  803 
9.66  827 

9.72  078 
9.72  109 
9.72  140 

0.27  922 
0.27  891 
0.27  860 

9.94  700 
9.94  694 
9.94  687 

16 
15 
14 

"\7    .6 

47 
48 
49 

9.66  851 
9.66  875 
9.66  899 

9.72  170 
9.72  201 
9.72  231 

0.27  830 
0.27  799 
0.27  769 

9.94  680 
9.94  674 
9.94  667 

13 
12 
11 
10 

9 

8 

7 

5  0.6  0.5 

I0'l.2  I.O 

I5;i.8  i.S 
20  2.312.0 
25!2.9  2.5 
30'3.5  3.0 
35:4.1  3.5 
404.714.0 
45  5.24.5 
50:5.8,5.0 

50 

9.66  922 

9.72  262 

0.27  738 

9.94  660 

51 

52 
53 

9.66  946 
9.66  970 
9.66  994 

9.72  293 
9.72  323 
9.72  354 

0.27  707 
0.27  677 
0.27  646 

9.94  654 
9.94  647 
9.94  640 

54 
55 
56 

9.67  018 
9.67  042 
9.67  066 

9.72  384 
9.72  415 
9.72  445 

0.27  616 
0.27  585 
0.27  555 

9.94  634 
9.94  627 
9.94  620 

6 

5 
4 

556.45.5 

57 
58 
59 

9.67  090 
9.67  113 
9.67  137 

9.72  476 
9.72  506 
9.72  537 
9.72  567 

0.27  524 
0.27  494 
0.27  463 

9.94  614 
9.94  607 
9.94  600 

3 
2 
1 
0 

60 

9.67  161 

0.27  433 

9.94  593 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

d. 

P.P. 

1 

152  28° 


30 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 


51 

52 
53 
54 
55 
56 
57 
58 
59 


60 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


9.67  161 


9.67  185 
9.67  208 
9.67  232 
9.67  256 
9.67  280 
9.67  303 
9.67  327 
9.67  350 
9.67  374 


9.67  398 


9.67  421 
9.67  445 
9.67  468 
9.67  492 
9.67  515 
9.67  539 
9.67  562 
9.67  586 
9.67  609 


9.67  633 


9.67  656 
9.67  680 
9.67  703 
9.67  726 
9.67  750 
9.67  773 
9.67  796 
9.67  820 
9.67  843 


9.67  866 


9.67  890 
9.67  913 
9.67  936 
9.67  959 

9.67  982 

9.68  006 
9.68  029 
9.68  052 
9.68  075 


9.68  098 


9.68  121 
9.68  144 
9.68  167 
9.68  190 
9.68  213 
9.68  237 
9.68  260 
9.68  283 
9.68  305 


9.68  328 


9.68  351 
9.68  374 
9.68  397 
9.68  420 
9.68  443 
9.68  466 
9.68  489 
9.68  512 
9.68  534 


9.68  557 
L.  Cos. 


24 
23 
24 
24 
24 
23 
24 
23 
24 
24 
23 
24 
23 
24 
23 
24 
23 
24 
23 
24 
23 
24 
23 
23 
24 
23 
23 
24 
23 
23 
24 
23 
23 
23 
23 
24 
23 
23 
23 
23 
23 
23 
23 
23 
23 
24 
23 
23 
22 
23 
23 
23 
23 
23 
23 
23 
23 
23 
22 
23 

d. 


9.72  567 


9.72  598 
9.72  628 
9.72  659 
9.72  689 
9.72  720 
9.72  750 
9.72  780 
9.72  811 
9.72  841 


9.72  872 


9.72  902 
9.72  932 
9.72  963 

9.72  993 

9.73  023 
9.73  054 
9.73  084 
9.73  114 
9.73  144 


9.73  175 


9.73  205 
9.73  235 
9.73  265 
9.73  295 
9.73  326 
9.73  356 
9.73  386 
9.73  416 
9.73  446 


9.73  476 


9.73  507 
9.73  537 
9.73  567 
9.73  597 
9.73  627 
9.73  657 
9.73  687 
9.73  717 
9.73  747 


9.73  777 


9.73  807 
9.73  837 
9.73  867 
9.73  897 
9.73  927 
9.73  957 

9.73  987 

9.74  017 
9.74  047 


9.74  077 


9.74  107 
9.74  137 
9.74  166 
9.74  196 
9.74  226 
9.74  256 
9.74  286 
9.74  316 
9.74  345 


9.74  375 
L.  Cot. 


31 
30 
31 
30 
31 
30 
30 
31 
30 
31 
30 
30 
31 
30 
30 
31 
30 
30 
30 
31 
30 
30 
30 
30 
31 
30 
30 
30 
30 
30 
31 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
30 
30 
30 
29 
30 

c.d. 


0.27  433 


0.27  402 
0.27  372 
0.27  341 
0.27  311 
0.27  280 
0.27  250 
0.27  220 
0.27  189 
0.27  159 


0.27  128 


9.27  098 
0.27  068 
0.27  037 
0.27  007 
0.26  977 
0.26  946 
0.26  916 
0.26  886 
0.26  856 


0.26  825 


0.26  795 
0.26  765 
0.26  735 
0.26  705 
0.26  674 
0.26  644 
0.26  614 
0.26  584 
0.26  554 


0.26  524 


0.26  493 
0.26  463 
0.26  433 
0.26  403 
0.26  373 
0.26  343 
0.26  313 
0.26  283 
0.26  253 


0.26  223 


0.26  193 
0.26  163 
0.26  133 
0.26  103 
0.26  073 
0.26  043 
0.26  013 
0.25  983 
0.25  953 


0.25  923 


0.25  893 
0.25  863 
0.25  834 
0.25  804 
0.25  774 
0.25  744 
0.25  714 
0.25  684 
0.25  655 


0.25  625 
L.  Tan. 


9.94  593 


9.94  587 
9.94  580 
9.94  573 
9.94  567 
9.94  560 
9.94  553 
9.94  546 
9.94  540 
9.94  533 


9.94  526 


9.94  519 
9.94  513 
9.94  506 
9.94  499 
9.94  492 
9.94  485 
9.94  479 
9.94  472 
9.94  465 


9.94  458 


9.94  451 
9.94  445 
9.94  438 
9.94  431 
9.94  424 
9.94  417 
9.94  410 
9.94  404 
9.94  397 


9.94  390 


9.94  383 
9.94  376 
9.94  369 
9.94  362 
9.94  355 
9.94  349 
9.94  342 
9.94  335 
9.94  328 


9.94  321 


9.94  314 
9.94  307 
9.94  300 
9.94  293 
9.94  286 
9.94  279 
9.94  273 
9.94  266 
9.94  259 


9.94  252 


9.94  245 
9.94  238 
9.94  231 
9.94  224 
9.94  217 
9.94  210 
9.94  203 
9.94  196 
9.94  189 


9.94  182 
L.  Sin. 


P.P. 


n 

31 

30 

29 

5 

2.6 

2.5 

2.4 

10 

5-2 

5.0 

4.8 

15 

7.8 

7-5 

7.2 

20  I0.3 

lO.O 

9.7 

2SjI2.9 

12.5 

12. 1 

30  IS.5 

iS-o 

14-5 

35  i8.i 

17.5 

16.9 

40  20.7  20.0  19.3 

45  23.2I22.S  21.8 

50  25.8  25.0  24.2 

55  28.4 

27.S  26.6 

24  I  23     22 


2.0    1.9 
4-0    3-8 

6.0  5.8 

8.01  7.7 
lo.o]  9.6 

3o!i2.o  11.5 
35  14.0  13-4 
40  16.0  15.3 
45  18.0  17.2 
SO  20.0  19.2 
55  22.0  21. 1 


1.8 
3-7 
5-5 
7-3 
9.2 

II.O 

12.8 
14.7 
16.S 
18.3 
20.2 


"I  7 

5i0.6  0.5 
lo;i.2  i.o 
151I.8  l.S 

20|2.3  2.0 
25|2.9'2.S 
303.53.0 

35'4.i!3.5 


4.74.0 
5.24.5 
5.8  5.0 


55  6.4  S.S 


P.P. 


29' 


153 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 

52 
53 
54 
55 
56 
57 
58 
59 


9.68  557 


9.68  580 
9.68  603 
9.68  625 
9.68  648 
9.68  671 
9.68  694 
9.68  716 
9.68  739 
9.68  762 


10  9.68  784 


60 


9.68  807 
9.68  829 
9.68  852 
9.68  875 
9.68  897 
9.68  920 
9.68  942 
9.68  965 
9.68  987 


9.69  010 


9.69  032 
9.69  055 
9.69  077 
9.69  100 
9.69  122 
9.69  144 
9.69  167 
9.69  189 
9.69  212 


9.69  234 


9.69  256 
9.69  279 
9.69  301 
9.69  323 
9.69  345 
9.69  368 
9.69  390 
9.69  412 
9.69  434 


9.69  456 


9.69  479 
9.69  501 
9.69  523 
9.69  545 
9.69  567 
9.69  589 
9.69  611 
9.69  633 
9.69  655 


9.69  677 


9.69  699 
9.69  721 
9.69  743 
9.69  765 
9.69  787 
9.69  809 

9.69  831 
9.69  853 
9.69  875 


9.69  897 


23 

23 

22 

23 

23 

23 

22 

23 

23 

22 

23 

22 

23 

23 

22 

23 

22 

23 

22 

23 

22 

23 

22 

23 

22 

22 

23 

22 

23 

22 

22 

23 

22 

22 

22 

23 

22 

22 

22 

22 

23 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 

22 


9.74  375 
9.74  405 
9.74  435 
9.74  465 
9.74  494 
9.74  524 
9.74  554 
9.74  583 
9.74  613 
9.74  643 


9.74  673 


9.74  702 
9.74  732 
9.74  762 
9.74  791 
9.74  821 
9.74  851 
9.74  880 
9.74  910 
9.74  939 


9.74  969 


9.74  998 

9.75  028 
9.75  058 
9.75  087 
9.75  117 
9.75  146 
9.75  176 
9.75  205 
9.75  235 


9.75  264 


9.75  294 
9.75  323 
9.75  353 
9.75  382 
9.75  411 
9.75  441 
9.75  470 
9.75  500 
9.75  529 


9.75  558 


9.75  588 
9.75  617 
9.75  647 
9.75  676 
9.75  705 
9.75  735 
9.75  764 
9.75  793 
9.75  822 


9.75  852 


9.75  881 
9.75  910 
9.75  939 

.75  969 
.75  998 
.76  027 
.76  056 
.76  086 


9.76  115 


9.76  144 


30 

30 

30 

29 

30 

30 

29 

30 

30 

30 

29 

30 

30 

29 

30 

30 

29 

30 

29 

30 

29 

30 

30 

29 

30 

29 

30 

29 

30 

29 

30 

29 

30 

29 

29 

30 

29 

30 

29 

29 

30 

29 

30 

29 

29 

30 

29 

29 

29 

30 

29 

29 

29 

30 

29 

29 

29 

30 

29 

29 


0.25  625 


0.25  595 
0.25  565 
0.25  535 
0.25  506 
0.25  476 
0.25  446 
0.25  417 
0.25  387 
0.25  357 


0.25  327 


0.25  298 
0.25  268 
0.25  238 
0.25  209 
0.25  179 
0.25  149 
0.25  120 
0.25  090 
0.25  061 


9.94  182 


9.94  175 
9.94  168 
9.94  161 
9.94  154 
9.94  147 
9.94  140 
9.94  133 
9.94  126 
9.94  119 


9.94  112 


0.25  031 


0.25  002 
0.24  972 
0.24  942 
0.24  913 
0.24  883 
0.24  854 
0.24  824 
0.24  795 
0.24  765 


0.24  736 


0.24  706 
0.24  677 
0.24  647 
0.24  618 
0.24  589 
0.24  559 
0.24  530 
0.24  500 
0.24  471 


0.24  442 


0.24  412 
0.24  383 
0.24  353 
0.24  324 
0.24  295 
0.24  265 
0.24  236 
0.24  207 
0.24  178 


0.24  148 


0.24  119 
0.24  090 
0.24  061 

0.24  031 
0.24  002 
0.23  973 
0.23  944 
0.23  914 
0.23  885 


0.23  856 


9.94  105 
9.94  098 
9.94  090 
9.94  083 
9.94  076 
9.94  069 
9.94  062 
9.94  055 
9.94  048 


9.94  041 


9.94  034 
9.94  027 
9.94  020 
9.94  012 
9.94  005 
9.93  998 
9.93  991 
9.93  984 
9.93  977 


9.93  970 


9.93  963 
9.93  955 
9.93  948 
9.93  941 
9.93  934 
9.93  927 
9.93  920 
9.93  912 
9.93  905 


9.93  898 


9.93  891 
9.93  884 
9.93  876 
9.93  869 
9.93  862 
9.93  855 
9.93  847 
9.93  840 
9.93  833 


9.93  826 


9.93  819 
9.93  811 
9.93  804 
9.93  797 
9.93  789 
9.93  782 
9.93  775 
9.93  768 
9.93  760 


9.93  753 


60 

59 

58 
57 
56 
55 
54 
53 
52 

11. 
_50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 

IL 
20^ 

19 
18 
17 
16 
15 
14 
13 
12 

n 

10 


P.P. 


30 

29 

2.5 

2.4 

5-0 

4.8 

7-5 

7.2 

lO.O 

9.7 

12.5 

12. 1 

IS-O 

14.5 

17.516.9 

20.0  19.3 

22.5  21.8 

25.0(24.2 

27.5 

26.6I 

23 

1.9 

3.8 

5.8 

7.7 

9.6 

11.5 

13.4 

15-3 

17.2 

19.2 

21.1 


II 

22 

8  1  7 

5 

1.8 

0.7,0.6 

10 

3-7 

1.31.2 

IS 

5.5  2.0:1.8 

20 

7.3  2.7|2.3 

25 

9.2 

3.32.9 

30 

II.O 

4.03.5 

35  12.8 

4.7  4.1 

40  14.7 

5.3  4.7 

45  16.5  6.0I5.2 

50  18.3,6.715.8 

55 

20.2 

7.316.4 

L.  Cos. 


d. 


L.  Cot. 


c.d. 


L.  Tan.   L.  Sin. 


d. 


p.p. 


154 


30* 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20^ 
21 
22 
23 
24 
25 
26 
27 
28 
29 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


9.69  897 


9.69  919 
9.69  941 
9.69  963 

9.69  984 

9.70  006 
9.70  028 
9.70  050 
9.70  072 
9.70  093 


9.70  115 


9.70  137 
9.70  159 
9.70  180 
9.70  202 
9.70  224 
9.70  245 
9.70  267 
9.70  288 
9.70  310 


9.70  332 


9.70  353 
9.70  375 
9.70  396 
9.70  418 
9.70  439 
9.70  461 
9.70  482 
9.70  504 
9.70  525 


9.70  547 


9.70  568 
9.70  590 
9.70  611 
9.70  633 
9.70  654 
9.70  675 
9.70  697 
9.70  718 
9.70  739 


9.70  761 


9.70  782 
9.70  803 
9.70  824 
9.70  846 
9.70  867 
9.70  888 
9.70  909 
9.70  931 
9.70  952 


9.70  973 


9.70  994 
9.71015 

9.71  036 
9.71  058 
9.71  079 
9.71  100 
9.71  121 
9.71  142 
9.71  163 


9.71  184 


L.  Cos. 


22 
22 
22 
21 
22 
22 
22 
22 
21 
22 
22 
22 
21 
22 
22 
21 
22 
21 
22 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
22 
21 
21 
22 
21 
21 
22 
21 
21 
21 
22 
21 
21 
21 
22 
21 
21 
21 
21 
21 
22 
21 
21 
21 
21 
21 
21 


L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


76  144 


76  173 
76  202 
76  231 
76  261 
76  290 
76  319 
76  348 
76  377 
76  406 


76  435 


76  464 
76  493 
76  522 
76  551 
76  580 
76  609 
76  639 
76  668 
76  697 


76  725 


76  754 
76  783 
76  812 
76  841 
76  870 
76  899 
76  928 
76  957 
76  986 


77  015 


77  044 
77  073 
77  101 
77  130 
77  159 
77  188 
77  217 
77  246 
77  274 


77  303 


77  332 
77  361 
77  390 
77  418 
77  447 
77  476 
77  505 
77  533 
77  562 


77  591 


77  619 
77  648 
77  677 
77  706 
77  734 
77  763 
77  791 
77  820 
77  849 


77  877 


d.   L.  Cot.  c.d 


29 

29 
29 
30 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
30 
29 
29 
28 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
28 
29 
29 
29 
29 
29 
28 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 
29 
29 
28 
29 
28 
29 
29 
28 


0.23  856 


0.23  827 
0.23  798 
0.23  769 
0.23  739 
0.23  710 
0.23  681 
0.23  652 
0.23  623 
0.23  594 


0.23  565 


0.23  536 
0.23  507 
0.23  478 
0.23  449 
0.23  420 
0.23  391 
0.23  361 
0.23  332 
0.23  303 


0.23  275 


0.23  246 
0.23  217 
0.23  188 
0.23  159 
0.23  130 
0.23  101 
0.23  072 
0.23  043 
0.23  014 


0.22  985 


0.22  956 
0.22  927 
0.22  899 
0.22  870 
0.22  841 
0.22  812 
0.22  783 
0.22  754 
0.22  726 


0.22  697 


0.22  668 
0.22  639 
0.22  610 
0.22  582 
0.22  553 
0.22  524 
0.22  495 
0.22  467 
0.22  438 


0.22  409 
0.22  381 
0.22  352 
0.22  323 
0.22  294 
0.22  266 
0.22  237 


0.22  209 
0.22  180 
0.22  151 


0.22  123 


9.93  753 


9.93  746 
9.93  738 
9.93  731 
9.93  724 
9.93  717 
9.93  709 
9.93  702 
9.93  695 
9.93  687 


9.93  680 


9.93  673 
9.93  665 
9.93  658 
9.93  650 
9.93  643 
9.93  636 
9.93  628 
9.93  621 
9.93  614 


9.93  606 


9.93  599 
9.93  591 
9.93  584 
9.93  577 
9.93  569 
9.93  562 
9.93  554 
9.93  547 
9.93  539 


9.93  532 


9.93  525 
9.93  517 
9.93  510 
9.93  502 
9.93  495 
9.93  487 
9.93  480 
9.93  472 
9.93  465 


9.93  457 


9.93  450 
9.93  442 
9.93  435 
9.93  427 
9.93  420 
9.93  412 
9.93  405 
9.93  397 
9.93  390 


9.93  382 


9.93  375 
9.93  367 
9.93  360 
9.93  352 
9.93  344 
9.93  337 
9.93  329 
9.93  322 
9.93  314 


9.93  307 


L.  Tan.     L.  Sin. 


d. 


P.P. 


30  I  29  I  28 


S  2.5 
lo]  5.0 
I5j   7-5 

20:10.0 


2.4 

4.8i 
7.2, 
9.7! 


2.3 

4.7 
7.0 
9.3 


25:12.5  12. I|II.7 
!i4-5  14-0 
■16.9  16.3 
19.3,18.7 

2I.8'2I.O 

24.2  23.3 
26.6  25.7 


30:15.0: 

3S|i7.5: 
40  20.0 
45,22.5' 
50  25.0 
5527.5 


22     21 


1.8 
3.5 
5-2 
7.0 


30  ii.o  lo.s 
35  12.8  12.2 


40  14.7 
45  16.5 
50  18.3 
55  20.2 


14.0 
15.8 
17.S 
19.2 


:i 


8,7 
50.70.6 
10 1.3 1.2 
1512.011.8 

20J2.7,2.3 

25  3-3  2.9 
4-03.5 
4.7|4.I 
S.34.7 
6.0  5.2 
S0i6.7  5.8 
557.36.4 


P.P. 


0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

11 
12 
13 
14 
15 
16 
17 
18 

21 

22 
23 
24 
25 
26 
27 
28 
29 

31 
32 
33 
34 
35 
36 
37 
38 

41 
42 
43 
44 
45 
46 
47 
48 

50 


51 
52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin. 


9.71  184 


9.71  205 
9.71  226 
9.71  247 
9.71  268 
9.71  289 
9.71310 
9.71331 
9.71  352 
9.71  373 


9.71  393 


9.71  414 
9.71  435 
9.71  456 
9.71  477 
9.71498 
9.71519 
9.71  539 
9.71  560 
9.71  581 


9.71  602 


9.71  622 
9.71  643 
9.71  664 
9.71  685 
9.71  705 
9.71  726 
9.71  747 
9.71  767 
9.71  788 


9.71  809 


71829 
71  850 
71870 
71891 
71911 
71932 
71952 
71973 
71994 


72  014 


9.72  034 
9.72  055 
9.72  075 
9.72  096 
9.72  116 
9.72  137 
9.72  157 
9.72  177 
9.72  198 


9.72  218 


72  238 
72  259 
72  279 
72  299 
72  320 
72  340 
72  360 
72  381 
72  401 


72  421 


L.  Cos. 


21 
21 
21 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
20 
21 
21 
21 
20 
21 
21 
20 
21 
21 
20 
21 
20 
21 
20 
21 
20 
21 
21 
20 
20 
21 
20 
21 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 


31' 


L.  Tan.    c.d.     L.  Cot.      L.  Cos 


155 


77  877 


77  906 
77  935 
77  963 

77  992 
78020 

78  049 
78  077 
78  106 
78  135 


78  163 


78  192 
78  220 
78  249 
78  277 
78  306 
78  334 
78  363 
78  391 
78  419 


78  448 


78  476 
78  505 
78  533 
78  562 
78  590 
78  618 
78  647 
78  675 
78  704 


78  732 


78  760 
78  789 
78  817 
78  845 
78  874 
78  902 
78  930 
78  959 
78  987 


79  015 


79  043 
79  072 
79  100 
79  128 
79  156 
79  185 
79  213 
79  241 
79  269 


79  297 


79  326 
79  354 
79  382 
79  410 
79  438 
79  466 
79  495 
79  523 
79  551 


79  579 


d.   L.  Cot.  c.d 


29 
29 

28 
29 
28 
29 
28 
29 
29 
28 
29 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
29 
28 
28 
28 


0.22  123 


0.22  094 
0.22  065 
0.22  037 
0.22  008 
0.21  980 
0.21951 
0.21  923 
0.21  894 
0.21  865 


0.21  837 


0.21  808 
0.21  780 
0.21  751 
0.21  723 
0.21  694 
0.21  666 
0.21  637 
0.21  609 
0.21  581 


0.21  552 


0.21  524 
0.21  495 
0.21  467 
0.21  438 
0.21  410 
0.21  382 
0.21  353 
0.21  325 
0.21  296 


0.21  268 


0.21  240 
0.21  211 
0.21  183 
0.21  155 
0.21  126 
0.21  098 
0.21  070 
0.21  041 
0.21  013 


0.20  985 


0.20  957 
0.20  928 
0.20  900 
0.20  872 
0.20  844 
0.20  815 
0.20  787 
0.20  759 
0.20  731 


0.20  703 


0.20  674 
0.20  646 
0.20  618 
0.20  590 
0.20  562 
0.20  534 
0.20  505 
0.20  477 
0.20  449 


0.20  421 


L.  Tan.   L.  Sin 


9^3^07 
9.93  299 
9.93  291 
9.93  284 
9.93  276 
9.93  269 
9.93  261 
9.93  253 
9.93  246 
9.93  238 


9.93  230 
9.93  223 
9.93  215 
9.93  207 
9.93  200 
9.93  192 
9.93  184 
9.93  177 
9.93  169 
9.93  161 


9.93  154 


9.93  146 
9.93  138 
9.93  131 
9.93  123 
9.93  115 
9.93  108 
9.93  100 
9.93  092 
9.93  084 


9.93  077 


9.93  069 
9.93  061 
9.93  053 
9.93  046 
9.93  038 
9.93  030 
9.93  022 
9.93  014 
9.93  007 


9.92  999 


9.92  991 
9.92  983 
9.92  976 
9.92  968 
9.92  960 
9.92  952 
9.92  944 
9.92  936 
9.92  929 


9.92  921 


9.92  913 
9.92  905 
9.92  897 
9.92  889 
9.92  881 
9.92  874 
9.92  866 
9.92  858 
9^2850 
9.92  842 


d. 


60 


59 

58 
57 
56 
55 
54 
53 
52 

11 
50^ 

49 

48 
47 
46 
45 
44 
43 
42 

11 
40 


P.P. 


"  I  29  i  28  I  21 


SI    2 

101  4 

IS,  7 
20  9 
25  12 
30  14 
35  i6 
40  19 
4S  21 
5024 
SS  26 


.4  2.3  1.8 
.8{  4.7I  3-5 
,21  7.0;  S.2 
7|  9.3]  7.0 
,1  11.71  8.8 
,5  14.010.5 
,9  16.3  12.2 
,3]  18.7  14-0 
,821.0  15.8 
2  23.3  17.S 
6125.7  19.2 


"    20  I  8  1  7 


1.7I0.7JO.6 

3.31.3  1.2 
5.0  2.0  1.8 

6.7 '2.7 '2.3 
8.3  3.3'2.9 
0.0  4.0  3-5 
35  ii.7  4.7|4-I 
40  13.3  5-3  4-7 
45  15.0  6.0  5.2 
50  16.7  6.7  5-8 
55  18.3  7-3  6.4 


P.P. 


156 


32° 


21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39^ 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


L.  Sin. 


9.72  421 


9.72  441 
9.72  461 
9.72  482 
9.72  502 
9.72  522 
9.72  542 
9.72  562 
9.72  582 
9.72  602 


9.72  622 


9.72  643 
9.72  663 
9.72  683 
9.72  703 
9.72  723 
9.72  743 
9.72  763 
9.72  783 
9.72  803 


9.72  823 


9.72  843 
9.72  863 
9.72  883 
9.72  902 
9.72  922 
9.72  942 
9.72  962 

9.72  982 

9.73  002 


9.73  022 


9.73  041 
9.73  061 
9.73  081 
9.73  101 
9.73  121 
9.73  140 
9.73  160 
9.73  180 
9.73  200 


9.73  219 


9.73  239 
9.73  259 
9.73  278 
9.73  298 
9.73  318 
9.73  337 
9.73  357 
9.73  377 
9.73  396 


73  416 


.73  435 
.73  455 
.73  474 
.73  494 
.73  513 
.73  533 
.73  552 
.73  572 
.73  591 


73  611 


20 
20 
21 
20 
20 
20 
20 
20 
20 
20 
21 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 
20 
19 
20 
20 
20 
19 
20 
20 
19 
20 
20 
19 
20 
20 
19 
20 
19 
20 
19 
20 
19 
20 
19 
20 
19 
20 


L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


9.79  579 


9.79  607 
9.79  635 
9.79  663 
9.79  691 
9.79  719 
9.79  747 
9.79  776 
9.79  804 
9.79  832 


9.79  860 


9.79  888 
9.79  916 
9.79  944 

9.79  972 

9.80  000 
9.80  028 
9.80  056 
9.80  084 
9.80  112 


9.80  140 


9.80  168 
9.80  195 
9.80  223 
9.80  251 
9.80  279 
9.80  307 
9.80  335 
9.80  363 
9.80  391 


9.80  419 


9.80  447 
9.80  474 
9.80  502 
9.80  530 
9.80  558 
9.80  586 
9.80  614 
9.80  642 
9.80  669 


9.80  697 


9.80  725 
9.80  753 
9.80  781 
9.80  808 
9.80  836 
9.80  864 
9.80  892 
9.80  919 
9.80  947 


9.80  975 


9.81  003 
9.81  030 
9.81  058 
9.81  086 
9.81  113 
9.81  141 
9.81  169 
9.81  196 
9.81  224 


9.81  252 


28 
28 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
27 
28 
28 
27 
28 
28 


0.20  421 


0.20  393 
0.20  365 
0.20  337 
0.20  309 
0.20  281 
0.20_253 
0.20  224 
0.20  196 
0.20  168 


0.20  140 


0.20  112 
0.20  084 
0.20  056 
0.20  028 
0.20  000 
0.19  972 
0.19  944 
0.19  916 
0.19  888 


0.19  860 


0.19  832 
0.19  805 
0.19  777 
0.19  749 
0.19  721 
0.19  693 
0.19  665 
0.19  637 
0.19  609 


0.19  581 


0.19  553 
0.19  526 
0.19  498 
0.19  470 
0.19  442 
0.19  414 
0.19  386 
0.19  358 
0.19  331 


0.19  303 


0.19  275 
0.19  247 
0.19  219 
0.19  192 
0.19  164 
0.19  136 
0.19  108 
0.19  081 
0.19  053 


0.19  025 


0.18  997 
0.18  970 
0.18  942 
0.18  914 
0.18  887 
0.18  859 
0.18  831 
0.18  804 
0.18  776 


0.18  748 


9.92  842 


9.92  834 
9.92  826 
9.92  818 
9.92  810 
9.92  803 
9.92  795 
9.92  787 
9.92  779 
9.92  771 


9.92  763 


9.92  755 
9.92  747 
9.92  739 
9.92  731 
9.92  723 
9.92  715 
9.92  707 
9.92  699 
9.92  691 


9.92  683 


9.92  675 
9.92  667 
9.92  659 
9.92  651 
9.92  643 
9.92  635 
9.92  627 
9.92  619 
9.92  611 


9.92  603 


9.92  595 
9.92  587 
9.92  579 
9.92  571 
9.92  563 
9.92  555 
9.92  546 
9.92  538 
9.92  530 


9.92  522 


9.92  514 
9.92  506 
9.92  498 
9.92  490 
9.92  482 
9.92  473 
9.92  465 
9.92  457 
9.92  449 


9.92  441 


9.92  433 
9.92  425 
9.92  416 
9.92  408 
9.92  400 
9.92  392 
9.92  384 
9.92  376 
9.92  367 


9.92  359 


30 


P.P. 


29     28     27 


2.4 
4.8 
7.2 
9-7 

12. 1 

14-5 


2.2 

4-5 
6.8 
9.0 


7  II.2 

ojia-s 


35  i6.9!i6.3ji5.* 
.7  i8.0 

.o'20.2 
.3'22.5 
.724.8 


40  19.31 

45  2i.8;2i. 

50  24.2j23. 

SS'26.6'2S. 


21     20     19 


1.7 
3-3 
5.0 
7.01  6.7 
8.81  8.3 
io.5|io.o 
12.2  11.7 
14.0;  13.3 
I5.8|I5.0 
I7.5ji6.7 


S5'i9.2'i8.3 


1.6 

3.2 

4.8 

6.3 

7.9 

9.5 

II. I 

12.7 

14.2 

IS.8 

17.4 


9  I  8 


5  0.8  o 
10  1.5  I 


2.2  2 
3-0  2 
3.83 

4-5  4 
5.2  4 
6.0  5 


45:6.8  6 
50:7.56 
55  8.2  7 


171 
70.6 

3  1.2 
0  1.8 
7  2.3 
3|2.9 
03.5 
74.1 
3|4-7 
05.2 
75.8 
3i6.4 


L.  Cos. 


d.      L.  Cot.     c.d.     L.  Tan.     L.  Sin.      d. 


P.P. 


33^ 


157 


' 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

9.73  611 

19 
20 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
19 
19 
20 
19 
19 
19 
19 
20 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
18 
19 
19 
19 
19 
19 
18 
19 
19 
19 
18 
19 

9.81  252 

27 
28 
28 
27 
28 
28 
27 
28 
27 
28 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 
28 

0.18  748 

9.92  359 

8 
8 

60 

59 

1 

9.73  630 

9.81  279 

0.18  721 

9.92  351 

2 

9.73  650 

9.81  307 

0.18  693 

9.92  343 

8 

58 

3 

9.73  669 

9.81  335 

0.18  665 

9.92  335 

9 

8 

57 

4 

9.73  689 

9.81  362 

0.18  638 

9.92  326 

56 

5 

9.73  708 

9.81  390 

0.18  610 

9.92  318 

8 

55 

6 

9.73  727 

9.81418 

0.18  582 

9.92  310 

8 

Q 

54 

"\   28 

27 

7 

9.73  747 

9.81  445 

0.18  555 

9.92  302 

53 

5  2.3 

2.2 

8 

9.73  766 

9.81  473 

0.18  527 

9.92  293 

8 

52 

10  4-7 
IS  7.0 
20  9-3 
25  II-7 
30  14-0 
35  i6.3 

4-5 
6  8 

9 

9.73  785 

9.81  500 

0.18  500 

9.92  285 

8 

8 
9 

51 
50 

49 

9.0 
II. 2 
13.S 
15.8 

10 

9.73  805 

9.81  528 

0.18  472 

9.92  277 

11 

9.73  824 

9.81  556 

0.18  444 

9.92  269 

12 

9.73  843 

9.81  583 

0.18  417 

9.92  260 

8 

48 

40  18.7 
45  21.0 
50  23.3 

18.0 

13 

9.73  863 

9.81611 

0.18  389 

9.92  252 

8 
9 

47 

22.5 

14 

9.73  882 

9.81  638 

0.18  362 

9.92  244 

46 

55  25.7 

24.8 

15 

9.73  901 

9.81  666 

0.18  334 

9.92  235 

8 

45 

16 

9.73  921 

9.81  693 

0.18  307 

9.92  227 

8 
8 
9 
8 
8 
9 
8 

44 

17 

9.73  940 

9.81  721 

0.18  279 

9.92  219 

43 

18 

9.73  959 

9.81  748 

0.18  252 

9.92  211 

42 

19 

9.73  978 

9.81  776 

0.18  224 

9.92  202 

41 
40 

39 

20 

9.73  997 

9.81  803 

0.18  197 

9.92  194 

21 

9.74  017 

9.81831 

0.18  169 

9.92  186 

22 

9.74  036 

9.81  858 

0.18  142 

9.92  177 

38 

23 

9.74  055 

9.81  886 

0.18  114 

9.92  169 

8 
9 

37 

24 

9.74  074 

9.81913 

0.18  087 

9.92  161 

36 

25 

9.74  093 

9.81  941 

0.18  059 

9.92  152 

8 
8 
9 
8 
8 
9 
8 
8 
9 
8 
9 
8 
8 
9 
8 
9 
8 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 

35 

26 

9.74  113 

9.81  968 

0.18  032 

9.92  144 

34 

" 

20 

19 

18 

27 

9.74  132 

9.81  996 

0.18  004 

9.92  136 

33 

5 

I.' 

1    1.6 

I.S 

28 

9.74  151 

9.82  023 

0.17  977 

9.92  127 

31 

10 

3-: 

5  3.2 
)  4.8 
1    6.3 
5  7.9 

3.0 

29 
30 

9.74  170 

9.82  051 

0.17  949 

9.92  119 

31 
30 

20 

25 
30 

6.' 

8.: 

TO  f 

6.0 
7.5 

9.74  189 

9.82  078 

0.17  922 

9.92  111 

31 

9.74  208 

9.82  106 

0.17  894 

9.92  102 

29 

35 

II.' 

1  II. I 

10.5 

32 

9.74  227 

9.82  133 

0.17  867 

9.92  094 

28 

40 

13.: 

J  12.7 

12.0 

33 

9.74  246 

9.82  161 

0.17  839 

9.92  086 

27 

50  16.' 

15.8 

15.0 

34 

9.74  265 

9.82  188 

0.17  812 

9.92  077 

26 

55  18.: 

H7.4 

16.5 

35 

9.74  284 

9.82  215 

0.17  785 

9.92  069 

25 

36 

9.74  303 

9.82  243 

0.17  757 

9.92  060 

24 

37 

9.74  322 

9.82  270 

0.17  730 

9.92  052 

23 

38 

9.74  341 

9.82  298 

0.17  702 

9.92  044 

22 

39 

9.74  360 

9.82  325 

0.17  675 

9.92  035 

21 
20 

40 

9.74  379 

9.82  352 

0.17  648 

9.92  027 

41 

9.74  398 

9.82  380 

0.17  620 

9.92  018 

19 

42 

9.74  417 

9.82  407 

0.17  593 

9.92  010 

18 

43 

9.74  436 

9.82  435 

0.17  565 

9.92  002 

17 

44 

9.74  455 

9.82  462 

0.17  538 

9.91  993 

16 

45 

9.74  474 

9.82  489 

0.17  511 

9.91  985 

15 

46 

9.74  493 

9.82  517 

0.17  483 

9.91  976 

14 

"    9 

8 

47 

9.74  512 

9.82  544 

0.17  456 

9.91  968 

13 

5  0.8 

0.7 

48 

9.74  531 

9.82  571 

0.17  429 

9.91  959 

12 

10 

i.S 

1.3 

49 
50 

9.74  549 

9.82  599 

0.17  401 

9.91  951 

11 
10 

20 
25 
30 

3-0 
3.8 

1  5 

2.7 
3-3 

4.0 

9.74  568 

9.82  626 

0.17  374 

9.91  942 

51 

9.74  587 

9.82  653 

0.17  347 

9.91  934 

9 

35 

5.2 

4-7 

52 

9.74  606 

9.82  681 

0.17  319 

9.91  925 

8 

40 

b.o 
5.8 
7.5 

1:5 

6.7 

53 

9.74  625 

9.82  708 

0.17  292 

9.91917 

7 

50 

54 

9.74  644 

9.82  735 

0.17  265 

9.91  908 

6 

55 

3.2  7.3 

55 

9.74  662 

9.82  762 

0.17  238 

9.91  900 

5 

56 

9.74  681 

9.82  790 

0.17  210 

9.91  891 

4 

57 

9.74  700 

9.82  817 

0.17  183 

9.91  883 

3 

-^ 

58 

9.74  719 

9.82  844 

0.17  156 

9.91  874 

2 

f 

!  59 

9.74  737 

9.82  871 

0.17  129 

9.91  866 

1 

60 

9.74  756 

9.82  899 

0.17  101 

9.91  857 

0 

1 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

56^ 


158 


34^ 


10 


20 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


51 
52 
53 
54 
55 
56 
57 
58 
59 


60 


L.  Sin. 


9.74  756 


9.74  775 
9.74  794 
9.74  812 
9.74  831 
9.74  850 
9.74  868 
9.74  887 
9.74  906 
9.74  924 


9.74  943 


9.74  961 
9.74  980 

9.74  999 

9.75  017 
9.75  036 
9.75  054 
9.75  073 
9.75  091 
9.75  110 


9.75  128 


75  147- 

75  165 

75  184 

75  202 

75  221 

9.75  239 

9.75  258 

9.75  276 

9.75  294 


9.75  313 


9.75  331 
9.75  350 
9.75  368 
9.75  386 
9.75  405 
9.75  423 
9.75  441 
9.75  459 
9.75  478 


9.75  496 


75  514 
75  533 
75  551 
75  569 
75  587 
75  605 


9.75  624 
9.75  642 
9.75  660 


9.75  678 


75  696 
75  714 
75  733 
75  751 
75  769 
75  787 
75  805 
75  823 
75  841 


9.75  859 


d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d 


19 
19 

18 
19 
19 
18 
19 
19 
18 
19 
18 
19 
19 
18 
19 
18 
19 
18 
19 
18 
19 
18 
19 
18 
19 
18 
19 
18 
18 
19 
18 
19 
18 
18 
19 
18 
18 
18 
19 
18 
18 
19 
18 
18 
18 
18 
19 
18 
18 
18 
18 
18 
19 
18 
18 
18 
18 
18 
18 
18 


9.82 
9.82 
9.82 
9.82 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 


899 


926 
953 
980 
008 
035 
062 
089 
117 
144 


9.83  171 


9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 


198 
225 
252 
280 
307 
334 
361 
388 
415 


9.83  442 


9.83 
9.83 
9.83 
9.83 


470 
497 
524 
551 
578 
605 
632 
659 
686 


9.83  713 


9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 
9.83 


740 
768 
795 
822 
849 
876 
903 
930 
957 


9.83  984 


9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 


Oil 
038 
065 
092 
119 
146 
173 
200 
227 


9.84  254 


9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 
9.84 


280 
307 
334 
361 
388 
415 
442 
469 
496 


9.84  523 


27 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 
27 
27 
28 
27 
27 
27 
27 
27 
27 
28 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
28 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
2'7 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
27 
27 
27 
27 
27 


0.17  101 


0.17  074 
0.17  047 
0.17  020 
0.16  992 
0.16  965 
0.16  938 
0.16  911 
0.16  883 
0.16  856 


0.16  829 
0.16  802 
0.16  775 
0.16  748 
0.16  720 
0.16  693 
0.16  666 
0.16  639 
0.16  612 
0.16  585 


0.16  558 


0.16  530 
0.16  503 
0.16  476 
0.16  449 
0.16  422 
0.16  395 
0.16  368 
0.16  341 
0.16  314 


0.16  287 


0.16  260 
0.16  232 
0.16  205 
0.16  178 
0.16  151 
0.16  124 
0.16  097 
0.16  070 
0.16  043 


0.16  016 


0.15  989 
0.15  962 
0.15  935 
0.15  908 
0.15  881 
0.15  854 
0.15  827 
0.15  800 
0.15  773 


0.15  746 


0.15  720 
0.15  693 
0.15  666 
0.15  639 
0.15  612 
0.15  585 
0.15  558 
0.15  531 
0.15  504 


0.15  477 


9.91  857 


9.91  849 
9.91  840 
9.91  832 
9.91  823 
9.91  815 
9.91  806 
9.91  798 
9.91  789 
9.91  781 


9.91  772 


9.91  763 
9.91  755 
9.91  746 
9.91  738 
9.91  729 
9.91  720 
9.91  712 
9.91  703 
9.91  695 


9.91  686 


9.91  677 
9.91  669 
9.91  660 
9.91  651 
9.91  643 
9.91  634 
9.91  625 
9.91  617 
9.91  608 


9.91  599 


9.91  591 
9.91  582 
9.91  573 
9.91  565 
9.91  556 
9.91  547 
9.91  538 
9.91  530 
9.91  521 


9.91  512 


9.91  504 
9.91  495 
9.91  486 
9.91  477 
9.91  469 
9.91  460 
9.91  451 
9.91  442 
9.91  433 


9.91  425 


9.91416 
9.91  407 
9.91  398 
9.91  389 
9.91  381 
9.91  372 
9.91  363 
9.91  354 
9.91  345 


9.91  336 


L.  Cos.      d.      L.  Cot.     c.d.     L.  Tan.     L.  Sin.  |   d 

55° 


P.P. 


28     27     26 


2.3  2.2 

4-7  4-5 

7.0  6.8 

93  9.0 


25  11.7,11.2 

3o!i4.o!i3.5 
3S'i6.3:iS.8 
40  i8.7ji8.o 
45  21.0  20.2 

S0J23.3:22.S 

5Sl2S.7'24.8i 


2.2 

4.3 

6.5 

8.7 

10.8 

13.0 

15.2 

17.3 

19.5 

21.7 

23.8 


19     18 


1.6 
3.2 
4.8 
6.3 
7.9 
9.5 
II. I 
12.7 
45  14.2 
50  15-8 
55  17-4 


1.5 

3.0 

4.S 

6.0 

7-5 

9.0 

10.5 

12.0 

13.S 

iS.o 

16.S 


9     8 


50.8 
ioii.5 
15,2.2 
203.0 
2S|3.8 
30|4.5 
35{S.2 
40  6.0 


.217-3 


P.P. 


35° 

159 

1 
1 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

9.75  859 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
17 

9.84  523 

27 
26 

0.15  477 

9.91  336 

8 
9 

60 

59 

1 

9.75  877 

9.84  550 

0.15  450 

9.91  328 

2 

9.75  895 

9.84  576 

27 
27 
27 
27 

0.15  424 

9.91  319 

Q 

58 

3 

9.75  913 

9.84  603 

0.15  397 

9.91  310 

9 
q 

57 

4 

9.75  931 

9.84  630 

0.15  370 

9.91  301 

56 

5 

9.75  949 

9.84  657 

0.15  343 

9.91  292 

9 

55 

6 

9.75  967 

9.84  684 

27 

97 

0.15  316 

9.91  283 

9 

8 
9 

54 

N 

27 

26 

7 

9.75  985 

9.84  711 

0.15  289 

9.91  274 

53 

5 

2.2 

2.2 

8 

9.76  003 

9.84  738 

26 

0.15  262 

9.91  266 

52 

lO 

4-5 
6  8 

i-l 

9 
10 

9.76  021 

9.84  764 

27 
27 
27 

0.15  236 

9.91  257 

9 
9 

Q 

51 
50 

49 

9.0  8.7      1 

9.76  039 

9.84  791 

0.15  209 

9.91  248 

25  II. 2  I0.8 

30,13-5  13-0 
35  IS.8, 15.2 

11 

9.76  057 

9.84  818 

0.15  182 

9.91  239 

12 

9.76  075 

9.84  845 

27 

0.15  155 

9.91  230 

9 
9 
9 
9 

9 
9 

48 

40  i8.o  17.3 
45  20.2  19.5 
50  22.5  21.7 

13 

9.76  093 

9.84  872 

27 

?6 

0.15  128 

9.91  221 

47 

14 

9.76  111 

9.84  899 

0.15  101 

9.91  212 

46 

Ssi24.8  23.8 

15 

9.76  129 

9.84  925 

97 

0.15  075 

9.91  203 

45 

16 

9.76  146 

18 
18 

9.84  952 

27 
27 

0.15  048 

9.91  194 

44 

17 

9.76  164 

9.84  979 

0.15  021 

9.91  185 

43 

18 

9.76  182 

18 

9.85  006 

27 

0.14  994 

9.91  176 

9 

42 

19 

9.76  200 

18 
18 
17 

9.85  033 

26 

27 
27 

0.14  967 

9.91  167 

9 
9 
8 
9 
9 
9 
9 

41 

20 

9.76  218 

9.85  059 

0.14  941 

9.91  158 

40 

39 

21 

9.76  236 

9.85  086 

0.14  914 

9.91  149 

22 

9.76  253 

18 

9.85  113 

27 

0.14  887 

9.91  141 

38 

23 

9.76  271 

18 
18 
17 

9.85  140 

26 

97 

0.14  860 

9.91  132 

37 

24 

9.76  289 

9.85  166 

0.14  834 

9.91  123 

36 

25 

9.76  307 

9.85  193 

27 

0.14  807 

9.91  114 

35 

26 

9.76  324 

18 
18 
18 

9.85  220 

27 
26 
27 

0.14  780 

9.91  105 

9 
9 
9 
9 
9 
9 
9 
9 

10 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

10 

34 

"  18  17 

27 

9.76  342 

9.85  247 

0.14  753 

9.91  096 

?>?^ 

s  1.5  1.4 

28 

9.76  360 

9.85  273 

0.14  727 

9.91  087 

32 

I5i  4-51  4-2 
20;  6.0  5.7 
25  7.5  7.1 
30:  9.0  8.5 
3510.5  9.9 

29 

9.76  378 

17 
18 
18 
17 

9.85  300 

27 
27 
26 
27 

0.14  700 

9.91  078 

31 
30 

29 

30 

9.76  395 

9.85  327 

0.14  673 

9.91  069 

31 

9.76  413 

9.85  354 

0.14  646 

9.91  060 

32 

9.76  431 

9.85  380 

0.14  620 

9.91  051 

28 

2>Z 

9.76  448 

18 
18 

9.85  407 

27 
26 

0.14  593 

9.91  042 

27 

5o]i5.o  14.2 

34 

9.76  466 

9.85  434 

0.14  566 

9.91  033 

26 

55  16.5  15.6 

35 

9.76  484 

17 

9.85  460 

27 

0.14  540 

9.91  023 

25 

36 

9.76  501 

18 
18 

9.85  487 

27 

96 

0.14  513 

9.91  014 

24 

37 

9.76  519 

9.85  514 

0.14  486 

9.91  005 

23 

38 

9.76  537 

17 

9.85  540 

27 

0.14  460 

9.90  996 

22 

39 

9.76  554 

18 

18 
17 

9.85  567 

27 

26 

97 

0.14  433 

9.90  987 

21 
20 

40 

9.76  572 

9.85  594 

0.14  406 

9.90  978 

41 

9.76  590 

9.85  620 

0.14  380 

9.90  969 

19 

42 

9.76  607 

18 

9.85  647 

97 

0.14  353 

9.90  960 

18 

43 

9.76  625 

17 

18 

9.85  674 

26 

97 

0.14  326 

9.90  951 

17 

44 

9.76  642 

9.85  700 

0.14  300 

9.90  942 

16 

45 

9.76  660 

17 

9.85  727 

97 

0.14  273 

9.90  933 

15 

46 

9.76  677 

18 
17 

18 

9.85  754 

26 

27 
97 

0.14  246 

9.90  924 

14 

"|10 

9  j  8 

47 

9.76  695 

9.85  780 

0.14  220 

9.90  915 

13 

5  0.8 

0.8  0.7 

48 

9.76  712 

9.85  807 

0.14  193 

9.90  906 

12 

10  1.7 
15  2.5 
20  3-3 
25  4-2 
30  5.0 

1.5  1.3 

49 
50 

9.76  730 

17 
18 
17 
18 

9.85  834 

26 

27 
26 

27 

0.14  166 

9.90  896 

9 
9 
9 
9 

11 
10 

3-0  2.7 
3-83.3 

4-5  4-0 

9.76  747 

9.85  860 

0.14  140 

9.90  887 

51 

9.76  765 

9.85  887 

0.14  113 

9.90  878 

9 

35  5.8 

5-2  4.7 

52 

9.76  782 

9.85  913 

0.14  087 

9.90  869 

8 

406.7 
45:7-5 
508.3 

6.0  5-3 
6860 

53 

9.76  800 

17 
18 

9.85  940 

27 
26 
27 

0.14  060 

9.90  860 

9 
9 
10 
9 
9 
9 
9 

7 

7-5  6.7 

54 

9.76  817 

9.85  967 

0.14  033 

9.90  851 

6 

55  9.218.2  7.3 

55 

9.76  835 

17 

9.85  993 

0.14  007 

9.90  842 

5 

56 

9.76  852 

18 
17 

9.86  020 

26 

27 

0.13  980 

9.90  832 

4 

57 

9.76  870 

9.86  046 

0.13  954 

9.90  823 

3 

1 

58 

9.76  887 

17 

9.86  073 

27 
26 

0.13  927 

9.90  814 

2 

59 

9.76  904 

18 

9.86  100 

0.13  900 

9.90  805 

1 
0 

» 

60 

9.76  922 

9.86  126 

0.13  874 

9.90  796 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

T.  Tan. 

L.  Sin. 

d. 

1 

P.P. 

54° 


160 

36° 

' 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

9.76  922 

17 
18 
17 
17 
18 
17 
17 
18 
17 
17 
17 
18 
17 
17 
17 
18 
17 
17 
17 
18 
17 
17 
17 
17 
17 
17 
17 
18 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
16 
17 
17 
17 
17 
17 
17 
16 
17 
17 
17 
17 
16 

9.86  126 

27 
26 
27 
26 
27 
26 
27 
26 
27 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 
27 
26 
26 
26 

0.13  874 

9.90  796 

9 

10 
9 
9 
9 
9 

10 
9 
9 
9 

10 
9 
9 
9 

10 
9 
9 
9 

10 
9 
9 

10 
9 
9 
9 

10 
9 
9 

'I 

9 
10 

9 
10 

9 

9 
10 

9 

9 
10 

9 
10 

9 
10 

9 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

9 
10 

9 

60 

59 

58 

57 

1 

2 
3 

9.76  939 
9.76  957 
9.76  974 

9.86  153 
9.86  179 
9.86  206 

0.13  847 
0.13  821 
0.13  794 

9.90  787 
9.90  777 
9.90  768 

4 
5 
6 

9.76  991 

9.77  009 
9.77  026 

9.86  232 
9.86  259 
9.86  285 

0.13  768 
0.13  741 
0.13  715 

9.90  759 
9.90  750 
9.90  741 

56 

55 
54 

// 

27  [  26 

7 
8 
9 

9.77  043 
9.77  061 
9.77  078 

9.86  312 
9.86  338 
9.86  365 

0.13  688 
0.13  662 
0.13  635 

9.90  731 
9.90  722 
9.90  713 

53 
52 
51 
50 

49 

48 
47 

5 
10 

15 

20 
25 
30 
35 

40 1 
451 
5oi 

2.2J  2.2 
4.5j  4.3 
6.8!  6.5 
9.0!  8.7 
11.2  10.8 
13.5  13.0 
15.8  15.2 
18.0  17.3 
20.2  19.5 
22..=;  21.7 

10 

9.77  095 

9.86  392 

0.13  608 

9.90  704 

11 
12 
13 

9.77  112 
9.77  130 
9.77  147 

9.86  418 
9.86  445 
9.86  471 

0.13  582 
0.13  555 
0.13  529 

9.90  694 
9.90  685 
9.90  676 

14 
15 
16 

9.77  164 
9.77  181 
9.77  199 

9.86  498 
9.86  524 
9.86  551 

0.13  502 
0.13  476 
0.13  449 

9.90  667 
9.90  657 
9.90  648 

46 
45 

44 

5si24.8  23.8 

17 
18 
19 

9.77  216 
9.77  233 
9.77  250 

9.86  577 
9.86  603 
9.86  630 

0.13  423 
0.13  397 
0.13  370 

9.90  639 
9.90  630 
9.90  620 

43 
42 
41 
40 

39 

38 
37 

20 

9.77  268 

9.86  656 

0.13  344 

9.90  611 

21 
22 
23 

9.77  285 
9.77  302 
9.77  319 

9.86  683 
9.86  709 
9.86  736 

0.13  317 
0.13  291 
0.13  264 

9.90  602 
9.90  592 
9.90  583 

24 
25 
26 

9.77  336 
9.77  353 
9.77  370 

9.86  762 
9.86  789 
9.86  815 

0.13  238 
0.13  211 
0.13  185 

9.90  574 
9.90  565 
9.90  555 

36 
35 
34 

It 

18  1  17 

16 

27 
28 
29 

9.77  387 
9.77  405 
9.77  422 

9.86  842 
9.86  868 
9.86  894 

0.13  158 
0.13  132 
0.13  106 

9.90  546 
9.90  537 
9.90  527 

33 
31 
31 
30 

29 

28 
27 

5 
10 

15 

20 

I 
3 
4 
6 

5  1.4 
0  2.8 
5  4.2 

0  5.7 
5  7.1 

01  8.5 

1.3 

2.7 
4.0 

11 

8.0 
9.3 
107 
12.0 
13.3 

30 

31 
32 
33 

9.77  439 

9.86  921 

0.13  079 

9.90  518 

25  7 
30!  9 

9.77  456 
9.77  473 
9.77  490 

9.86  947 

9.86  974 

9.87  000 

0.13  053 
0.13  026 
0.13  000 

9.90  509 
9.90  499 
9.90  490 

35  10.5I  9.9 
40  12.0  11.3 
45  13.5  12.8 
50  15.0  14.2 

34 
35 
36 

9.77  507 
9.77  524 
9.77  541 

9.87  027 
9.87  053 
9.87  079 

0.12  973 
0.12  947 
0.12  921 

9.90  480 
9.90  471 
9.90  462 

26 
25 
24 

55  16.S  IS.6  14.7 

37 
38 
39 

9.77  558 
9.77  575 
9.77  592 

9.87  106 
9.87  132 
9.87  158 

0.12  894 
0.12  868 
0.12  842 

9.90  452 
9.90  443 
9.90  434 

23 
22 
21 
20 
19 
18 
17 

i 

40 

9.77  609 

9.87  185 

0.12  815 

9.90  424 

41 

42 
43 

9.77  626 
9.77  643 
9.77  660 

9.87  211 
9.87  238 
9.87  264 

0.12  789 
0.12  762 
0.12  736 

9.90  415 
9.90  405 
9.90  396 

44 
45 
46 

9.77  677 
9.77  694 
9.77  711 

9.87  290 
9.87  317 
9.87  343 

0.12  710 
0.12  683 
0.12  657 

9.90  386 
9.90  377 
9.90  368 

16 
15 
14 

"  j  10|  9       1 

47 
48 
49 
50 
51 
52 
53 

9.77  728 
9.77  744 
9.77  761 
9.77  778 

9.87  369 
9.87  396 
9.87  422 

0.12  631 
0.12  604 
0.12  578 

9.90  358 
9.90  349 
9.90  339 

13 
12 
11 
10 

9 

8 

7 

5  0.8 
10  1.7 
152.5 
203.3 

25  4-2 

30  5-0 

0.8      1 

i.S 

2.2 

3.0 

3-8 

4.5 

5.2 

6.0 

6.8 

7.5 

9.87  448 

0.12  552 

9.90  330 

9.77  795 
9.77  812 
9.77  829 

9.87  475 
9.87  501 
9.87  527 

0.12  525 
0.12  499 
0.12  473 

9.90  320 
9.90  311 
9.90  301 

35 

40 
45 
50 

5.8 
6.7 
7.5 
8.3 

54 

55 
56 

9.77  846 
9.77  862 
9.77  879 

9.87  554 
9.87  580 
9.87  606 

0.12  446 
0.12  420 
0.12  394 

9.90  292 
9.90  282 
9.90  273 

6 

5 
4 

55  9.218.2 

57 
58 
59 
60 

9.77  896 
9.77  913 
9.77  930 

9.87  633 
9.87  659 
9.87  685 

0.12  367 
0.12  341 
0.12  315 

9.90  263 
9.90  254 
9.90  244 

3 
2 
1 
0 

9.77  946 

9.87  711 

0.12  289 

9.90  235 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan.   L.  Sin. 

d. 

' 

P.P. 

53° 


37° 


161 


10 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20^ 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


60 


L.  Sin.   d.   L.  Tan.  c.d.  !  L.  Cot.   L.  Cos.   d 


9.77  946 


9.77  963 
9.77  980 
9.77  997 
.78  013 
.78  030 
.78  047 
.78  063 
.78  080 
.78  097 


9.78  113 


9.78  130 
9.78  147 
9.78  163 
9.78  180 
9.78  197 
9.78  213 
9.78  230 
9.78  246 
9.78  263 


9.78  280 


9.78  296 
9.78  313 
9.78  329 
9.78  346 
9.78  362 
9.78  379 
9.78  395 
9.78  412 
9.78  428 


9.78  445 


9.78  461 
9.78  478 
9.78  494 
9.78  510 
9.78  527 
9.78  543 
9.78  560 
9.78  576 
9.78  592 


9.78  609 


9.78  625 
9.78  642 
9.78  658 
9.78  674 
9.78  691 
9.78  707 
9.78  723 
9.78  739 
9.78  756 


9.78  772 


9.78  788 
9.78  805 
9.78  821 
9.78  837 
9.78  853 
9.78  869 
9.78  886 
9.78  902 
9.78  918 


9.78  934 


9.87  711 


9.87  738 
9.87  764 
9.87  790 
9.87  817 
9.87  843 
9.87  869 
9.87  895 
9.87  922 
9.87  948 


9.87  974 


9.88  000 
9.88  027 
9.88  053 
9.88  079 
9.88  105 
9.88  131 
9.88  158 
9.88  184 
9.88  210 


9.88  236 


9.88  262 
9.88  289 
9.88  315 
9.88  341 
9.88  367 
9.88  393 
9.88  420 
9.88  446 
9.88  472 


9.88  498 


9.88  524 
9.88  550 
9.88  577 
9.88  603 
9.88  629 
9.88  655 
9.88  681 
9.88  707 
9.88  733 


9.88  759 


9.88  786 
9.88  812 
9.88  838 
9.88  864 
9.88  890 
9.88  916 
9.88  942 
9.88  968 
9.88  994 


9.89  020 


9.89  046 
9.89  073 
9.89  099 
9.89  125 
9.89  151 
9.89  177 
9.89  203 
9.89  229 
9.89  255 


9.89  281 


27 
26 
26 
27 
26 
26 
26 
27 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 


0.12  289 


0.12  262 
0.12  236 
0.12  210 
0.12  183 
0.12  157 
0.12  131 
0.12  105 
0.12  078 
0.12  052 


0.12  026 


0.12  000 
0.11973 
0.11  947 
0.11921 
0.11895 
0.11  869 
0.11  842 
0.11816 
0.11  790 


0.11  764 


0.11  738 
0.11  711 
0.11  685 
0.11  659 
0.11  633 
0.11  607 
0.11  580 
0.11554 
0.11  528 


0.11  502 


0.11476 
0.11  450 
0.11423 
0.11397 
0.11  371 
0.11  345 
0.11319 
0.11  293 
0.11  267 


0.11  241 


0.11  214 
0.11  188 
0.11  162 
0.11  136 
0.11  110 
0.11  084 

0.11058 
0.11032 
0.11006 


0.10  980 


0.10  954 
0.10  927 
0.10  901 
0.10  875 
0.10  849 
0.10  823 
0.10  797 
0.10  771 
0.10  745 


0.10  719 


9.90  235 


9.90  225 
9.90  216 
9.90  206 
9.90  197 
9.90  187 
9.90  178 
9.90  168 
9.90  159 
9.90  149 


9.90  139 


9.90  130 
9.90  120 
9.90  111 
9.90  101 
9.90  091 
9.90  082 
9.90  072 
9.90  063 
9.90  053 


9.90  043 


9.90  034 
9.90  024 
9.90  014 
9.90  005 
9.89  995 
9.89  985 
9.89  976 
9.89  966 
9.89  956 


9.89  947 


9.89  937 
9.89  927 
9.89  918 
9.89  908 
9.89  898 
9.89  888 
9.89  879 
9.89  869 
9.89  859 


9.89  849 


9.89  840 
9.89  830 
9.89  820 
9.89  810 
9.89  801 
9.89  791 
9.89  781 
9.89  771 
9.89  761 


9.89  752 


9.89  742 
9.89  732 
9.89  722 
9.89  712 
9.89  702 
9.89  693 
9.89  683 
9.89  673 
9.89  663 


9.89  653 


10 

9 
10 

9 
10 

9 
10 

9 
10 
10 

9 
10 

9 
10 
10 

9 
10 

9 
10 
10 

9 
10 
10 

9 
10 
10 

9 
10 
10 

9 
10 
10 

9 
10 
10 
10 

9 
10 
10 
10 

9 
10 
10 
10 

9 
10 
10 
10 
10 

9 
10 
10 
10 
10 
10 

9 
10 
10 
10 
10 


60^ 

59 

58 
57 
56 
55 
54 
53 
52 

11 
50 

49 

48 
47 
46 
45 
44 
43 
42 

il 
40 

39 
38 
37 
36 
35 
34 
S3 
32 
31 
_30 
29 
28 
27 
26 
25 
24 
23 
22 

20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
JO 
9 


P.P. 


27 

26 

2.2 

2.2 

4.5 

4..3 

6.8 

6.5 

Q.O 

8.7 

II.2 

10.8 

13.5 

13.0 

i5-8'i5-2' 

18.0  17.3 

20.2  19.5 

22.5  21.7 

24.8 

23.8I 

17 

1.4 

2.8 

4.2 

5.7 

7.1 

8.5 

9-9 

II.3 

12.8 

14.2 

15.6 


16  I  10|  9 


10| 


1.30.8  0.8 
2.7ii-7  1.5 
4.0  2.5  2.2 
5.3!3.3  3-0 
6.7'4.2  3-8 
8.0  5.0  4.5 
9-3  5.8  5-2 
40J10.7  6.7  6.0 
45  12.0  7.5  6.8 
50;  13.3  8.3  7-5 
55:14.79.2  8.2 


L.  Cos. 


d.       L.  Cot.     c.d.     L.  Tan.      L.  Sin, 


P.P. 


11 


52* 


162 


38° 


10 


20 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


9.78  934 


9.78  950 
9.78  967 
9.78  983 

9.78  999 

9.79  015 
9.79  031 
9.79  047 
9.79  063 
9.79  079 


9.79  095 


9.79  111 
9.79  128 
9.79  144 
9.79  160 
9.79  176 
9.79  192 
9.79  208 
9.79  224 
9.79  240 


9.79  256 


9.79  272 
9.79  288 
9.79  304 
9.79  319 
9.79  335 
9.79  351 
9.79  367 
9.79  383 
9.79  399 


9.79  415 


9.79  431 
9.79  447 
9.79  463 
9.79  478 
9.79  494 
9.79  510 
9.79  526 
9.79  542 
9.79  558 


9.79  573 


9.79  589 
9.79  605 
9.79  621 
9.79  636 
9.79  652 
9.79  668 
9.79  684 
9.79  699 
9.79  715 


79  731 


.79  746 
.79  762 
.79  778 
.79  793 
.79  809 
.79  825 
.79  840 
.79  856 
.79  872 


79  887 


L.  Cos. 


d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.  d. 


16 
17 
16 
16 
16 
16 
16 
16 
16 
16 
16 
17 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
15 
16 
16 
16 
16 
16 
16 
16 
16 
16 
15 
16 
16 
16 
16 
16 
15 
16 
16 
16 
15 
16 
16 
16 
15 
16 
16 
15 
16 
16 
15 
16 
16 
15 
16 
16 
15 


9.89  281 


9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 


307 
333 
359 
385 
411 
437 
463 
489 
515 


9.89  541 


9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 


567 
593 
619 
645 
671 
697 
723 
749 
775 


9.89  801 


9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.89 
9.90 
9.90 


827 
853 
879 
905 
931 
957 
983 
009 
035 


9.90  061 


9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 


086 
112 
138 
164 
190 
216 
242 
268 
294 


9.90  320 


9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 


346 
371 
397 
423 
449 
475 
501 
527 
553 


9.90  578 


9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 
9.90 


9.90 


604 
630 
656 
682 
708 
734 
759 
785 
811^ 
837 


L.  Cot.  c.d 


26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 


0.10  719 


0.10  693 
0.10  667 
0.10  641 
0.10  615 
0.10  589 
0.10  563 
0.10  537 
0.10  511 
0.10  485 


0.10  459 


0.10  433 
0.10  407 
0.10  381 
0.10  355 
0.10  329 
0.10  303 
0.10  277 
0.10  251 
0.10  225 


0.10  199 


0.10  173 
0.10  147 
0.10  121 
0.10  095 
0.10  069 
0.10  043 
0.10  017 
0.09  991 
0.09  965 


0.09  939 


0.09  914 
0.09  888 
0.09  862 
0.09  836 
0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 


0.09  680 


0.09  654 
0.09  629 
0.09  603 
0.09  577 
0.09  551 
0.09  525 
0.09  499 
0.09  473 
0.09  447 


0.09  422 


0.09  396 
0.09  370 
0.09  344 
0.09  318 
0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 


0.09  163 


L.  Tan. 


9.89  653 


9.89  643 
9.89  633 
9.89  624 
9.89  614 
9.89  604 
9.89  594 
9.89  584 
9.89  574 
9.89  564 


9.89  554 


9.89  544 
9.89  534 
9.89  524 
9.89  514 
9.89  504 
9.89  495 
9.89  485 
9.89  475 
9.89  465 


9.89  455 


9.89  445 
9.89  435 
9.89  425 
9.89  415 
9.89  405 
9.89  395 
9.89  385 
9.89  375 
9.89  364 


9.89  354 


9.89  344 
9.89  334 
9.89  324 
9.89  314 
9.89  304 
9.89  294 
9.89  284 
9.89  274 
9.89  264 


9.89  254 


9.89  244 
9.89  233 
9.89  223 
9.89  213 
9.89  203 
9.89  193 
9.89  183 
9.89  173 
9.89  162 


9.89  152 


9.89  142 
9.89  132 
9.89  122 
9.89  112 
9.89  101 
9.89  091 
9.89  081 
9.89  071 
9.89  060 


9.89  050 


L.  Sin. 


d. 


10 


P.P. 


26     25 


2.2 

4-3 

6.5 

8.7 

10.8 

I3-0 

15-2 

17.3 
19-5 
21.7 
23.8 


2.1 

4.2 
6.2 

8.3 

10.4 

12.5 

14.6 

16.7 
18.8 

20.8 

22.9 


17 

16 

1.4 

1.3 

2.8 

2.7 

4.2 

4.0 

S.7 

."5.3 

7.1 

6.7 

8.5 

8.0 

9.9 

9.3 

11.3 

10.7 

12.8 

12.0 

14.2 

13-3 

1S.6 

14.7 

1.2 

2.5 

3.8 
50 

6.2 

7.5 

8.8 

lO.O 

II. 2 
12.S 
13.8 


11     10   9 


0.9  0.8 
1.8  1.7 
2.8  2.5 


3-7 
4.6 
5-5 
6.4 
7-3 
8.2 
9.2 
10. 1 


P.P. 


51° 


39° 


163 


/ 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

"o" 

1 

9.79  887 

16 
15 
16 
16 
15 
16 
15 
16 

9.90  837 

26 
26 
25 
26 
26 
?6 

0.09  163 

9.89  050 

10 
10 

60 

59 

9.79  903 

9.90  863 

0.09  137 

9.89  040 

2 

9.79  918 

9.90  889 

0.09  111 

9.89  030 

10 

58 

3 

9.79  934 

9.90  914 

0.09  086 

9.89  020 

11 

10 

57 

4 

9.79  950 

9.90  940 

0.09  060 

9.89  009 

56 

5 

9.79  965 

9.90  966 

0.09  034 

9.88  999 

10 

55 

6 

9.79  981 

9.90  992 

26 
25 

0.09  008 

9.88  989 

11 

10 

54 

7 

9.79  996 

9.91  018 

0.08  982 

9.88  978 

53 

8 

9.80  012 

15 

16 

15 
16 

9.91  043 

26 
26 
26 
26 
25 
26 
26 

0.08  957 

9.88  968 

10 

52 

9 

9.80  027 

9.91  069 

0.08  931 

9.88  958 

10 

11 

in 

51 
50 

49 

"  5>R  1  5»i;  1  ie 

10 

9.80  043 

9.91  095 

0.08  905 

9.88  948 

11 

9.80  058 

9.91  121 

0.08  879 

9.88  937 

12 

9.80  074 

15 
16 
15 

9.91  147 

0.08  853 

9.88  927 

10 

48 

5 
10 

2  2 

2  I 

1.3 

2.7 

13 

9.80  089 

9.91  172 

0.08  828 

9.88  917 

11 

10 

47 

4-3 

4.2 

14 

9.80  105 

9.91  198 

0.08  802 

9.88  906 

46 

15 

0.5 

8  7 

b.2 

8.3 

10.4 

4.0 

1:? 

15 

9.80  120 

16 

9.91  224 

26 

0.08  776 

9.88  896 

10 

45 

25 

10.8 

16 

9.80  136 

15 
1  s 

9.91  250 

26 

95 

0.08  750 

9.88  886 

11 

10 

44 

30 

13.0 

12.S 

8.0 

17 

9.80  151 

9.91  276 

0.08  724 

9.88  875 

43 

35 

40 

17.3  16.7 

9.3 
10.7 

18 

9.80  166 

16 
15 
16 
15 

9.91  301 

26 
26 
26 
25 
26 

0.08  699 

9.88  865 

10 

11 

10 
10 

42 

45 

19.5 

18.8 

12.0 

19 
20 

9.80  182 
9.80  197 

9.91  327 

0.08  673 

9.88  855 

41 
40 

39 

50 
55' 

21.7 
2^.8 

20.8 
22.0 

13.3 

1A.7 

9.91  353 

0.08  647 

9.88  844 

21 

9.80  213 

9.91  379 

0.08  621 

9.88  834 

22 

9.80  228 

16 

9.91  404 

0.08  596 

9.88  824 

11 

38 

23 

9.80  244 

15 
15 
16 
15 

1  !? 

9.91  430 

26 
26 

25 

26 
?6 

0.08  570 

9.88  813 

10 
10 

37 

24 

9.80  259 

9.91  456 

0.08  544 

9.88  803 

36 

25 

9.80  274 

9.91  482 

0.08  518 

9.88  793 

11 

10 
1 1 

35 

26 

9.80  290 

9.91  507 

0.08  493 

9.88  782 

34 

27 

9.80  305 

9.91  533 

0.08  467 

9.88  772 

33 

28 

9.80  320 

16 

9.91  559 

96 

0.08  441 

9.88  761 

10 

32 

29 

9.80  336 

15 
15 
16 

9.91  585 

25 
26 
26 

0.08  415 

9.88  751 

10 

11 

10 

31 

30 

9.80  351 

9.91  610 

0.08  390 

9.88  741 

30 

31 

9.80  366 

9.91  636 

0.08  364 

9.88  730 

29 

32 

9.80  382 

15 

15 
16 

9.91  662 

26 

25 
96 

0.08  338 

9.88  720 

11 

28 

33 

9.80  397 

9.91  688 

0.08  312 

9.88  709 

10 
1 1 

27 

34 

9.80  412 

9.91  713 

0.08  287 

9.88  699 

26 

35 

9.80  428 

15 
15 
15 

16 

9.91  739 

26 
26 
25 
76 

0.08  261 

9.88  688 

10 
10 

11 

10 

11 

10 
1 1 

25 

36 

9.80  443 

9.91  765 

0.08  235 

9.88  678 

24 

37 

9.80  458 

9.91  791 

0.08  209 

9.88  668 

23 

38 

9.80  473 

9.91  816 

0.08  184 

9.88  657 

22 

39 
40 

9.80  489 

15 

15 
1 1; 

9.91  842 

26 

25 
96 

0.08  158 

9.88  647 

21 
20 

19 

9.80  504 

9.91  868 

0.08  132 

9.88  636 

41 

9.80  519 

9.91  893 

0.08  107 

9.88  626 

e 

I  2 

0  0 

0  8 

42 

9.80  534 

16 

9.91  919 

26 

26 

95 

0.08  081 

9.88  615 

10 

11 
in 

18 

10 

2.5 

1.8 

1.7 

43 

9.80  550 

15 
1  s 

9.91  945 

0.08  055 

9.88  605 

17 

15 

3.8 

2.8 

11 

44 

9.80  565 

9.91  971 

0.08  029 

9.88  594 

16 

25 

i).0 

6.2 

3-7 
4.6 

3.3 

4.2 

45 

9.80  580 

15 
15 
15 

16 

9.91.996 

26 
26 
25 
96 

0.08  004 

9.88  584 

11 

10 

11 

10 

11 

10 

11 
11 

10 

11 

10 

11 

10 

11 
11 

15 

30 

7.5 

5.5 

5.0 

46 

9.80  595 

9.92  022 

0.07  978 

9.88  573 

14 

35'  0.0 
40  lo.o 

0.4 
7.3 

5.8 
6.7 

47 

9.80  610 

9.92  048 

0.07  952 

9.88  563 

13 

45  II.2 

8.2 

7.S 

48 

9.80  625 

9.92  073 

0.07  927 

9.88  552 

12 

SO  I2.S 

y.2 

m   T 

8.3 

n  0 

49 

9.80  641 

15 
15 

9.92  099 

26 
25 
26 
26 
25 
26 
26 

0.07  901 

9.88  542 

11 
10 

50 

9.80  656 

9.92  125 

0.07  875 

9.88  531 

51 

9.80  671 

9.92  150 

0.07  850 

9.88  521 

9 

52 

9.80  686 

9.92  176 

0.07  824 

9.88  510 

8 

53 

9.80  701 

15 
15 

9.92  202 

0.07  798 

9.88  499 

7 

54 

9.80  716 

9.92  227 

0.07  773 

9.88  489 

6 

55 

9.80  731 

15 

9.92  253 

0.07  747 

9.88  478 

5 

56 

9.80  746 

16 
15 

9.92  279 

25 
26 

0.07  721 

9.88  468 

4 

57 

9.80  762 

9.92  304 

0.07  696 

9.88  457 

3 

58 

9.80  777 

15 
15 

9.92  330 

26 

25 

0.07  670 

9.88  447 

2 

59 

9.80  792 

9.92  356 

0.07  644 

9.88  436 

1 
0 

60 

9.80  807 

9.92  381 

0.07  619 

9.88  425 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

d. 

/ 

P.P. 

50^ 


164 


40° 


1 

2 
3 
4 
5 
6 
7 
8 
9 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


L.  Sin.   d.   L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


9.80  807 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59^ 
60 


9.80  822 
9.80  837 
9.80  852 
9.80  867 
9.80  882 
9.80  897 
9.80  912 
9.80  927 
9.80  942 


9.80  957 


9.80  972 

9.80  987 

9.81  002 
9.81  017 
9.81  032 
9.81  047 
9.81  061 
9.81  076 
9.81  091 


9.81  106 


9.81  121 
9.81  136 
9.81  151 
9.81  166 
9.81  180 
9.81  195 
9.81  210 
9.81  225 
9.81  240 
9.81  254 


9.81  269 
9.81  284 
9.81  299 
9.81  314 
9.81  328 
9.81  343 
9.81  358 
9.81  372 
9.81  387 


9.81  402 


9.81417 
9.81  431 
9.81  446 
9.81  461 
9.81  475 
9.81  490 
9.81  505 
9.81  519 
9.81  534 


9.81  549 


9.81  563 
9.81  578 
9.81  592 
9.81  607 
9.81  622 
9.81  636 
9.81  651 
9.81  665 
9.81  680 


9.81  694 


9.92  381 


9.92  407 
9.92  433 
9.92  458 
9.92  484 
9.92  510 
9.92  535 
9.92  561 
9.92  587 
9.92  612 


9.92  638 


9.92  663 
9.92  689 
9.92  715 
9.92  740 
9.92  766 
9.92  792 
9.92  817 
9.92  843 
9.92  868 


9.92  894 


9.92  920 
9.92  945 
9.92  971 

9.92  996 

9.93  022 
9.93  048 
9.93  073 
9.93  099 
9.93  124 


9.93  150 


9.93  175 
9.93  201 
9.93  227 
9.93  252 
9.93  278 
9.93  303 
9.93  329 
9.93  354 
9.93  380 


9.93  406 


9.93  431 
9.93  457 
9.93  482 
9.93  508 
9.93  533 
9.93  559 
9.93  584 
9.93  610 
9.93  636 


9.93  661 


9.93  687 
9.93  712 
9.93  738 
9.93  763 
9.93  789 
9.93  814 
9.93  840 
9.93  865 
9.93  891 


9.93  916 


26 

26 

25 

26 

26 

25 

26 

26 

25 

26 

25 

26 

26 

25 

26 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 

25 

26 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 

26 

25 


0.07  619 


0.07  593 
0.07  567 
0.07  542 
0.07  516 
0.07  490 
0.07  465 
0.07  439 
0.07  413 
0.07  388 


0.07  362 


0.07  337 
0.07  311 
0.07  285 
0.07  260 
0.07  234 
0.07  208 
0.07  183 
0.07  157 
0.07  132 


0.07  106 


0.07  080 
0.07  055 
0.07  029 
0.07  004 
0.06  978 
0.06  952 
0.06  927 
0.06  901 
0.06  876 


9.88  425 


9.88  415 
9.88  404 
9.88  394 
9.88  383 
9.88  372 
9.88  362 
9.88  351 
9.88  340 
9.88  330 
9.88  319 


9.88  308 
9.88  298 
9.88  287 
9.88  276 
9.88  266 
9.88  255 
9.88  244 
9.88  234 
9.88  223 


9.88  212 


0.06  850 


0.06  825 
0.06  799 
0.06  77:3 
0.06  748 
0.06  722 
0.06  697 
0.06  671 
0.06  646 
0.06  620 


0.06  594 


0.06  569 
0.06  543 
0.06  518 
0.06  492 
0.06  467 
0.06  441 
0.06  416 
0.06  390 
0.06  364 


0.06  339 


0.06  313 
0.06  288 
0.06  262 
0.06  237 
0.06  211 
0.06  186 
0.06  160 
0.06  135 
0.06  109 


0.06  084 


9.88  201 
9.88  191 
9.88  180 
9.88  169 
9.88  158 
9.88  148 
9.88  137 
9.88  126 
9.88  115 


9.88  105 


9.88  094 
9.88  083 
9.88  072 
9.88  061 
9.88  051 
9.88  040 
9.88  029 
9.88  018 
9.88  007 


9.87  996 
9.87  985 
9.87  975 
9.87  964 
9.87  953 
9.87  942 
9.87  931 
9.87  920 
9.87  909 
9.87  898 


9.87  887 


9.87  877 
9.87  866 
9.87  855 
9.87  844 
9.87  833 
9.87  822 
9.87  811 
9.87  800 
9.87  789 


9.87  778 


L.  Cos.   d.   L.  Cot.  c.d.  L.  Tan.   L.  Sin.   d. 

49° 


60 

59 
58 
57 
56 
55 
54 
53 
52 
_M 
_50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 

iL 
30^ 

29 
28 
27 
26 
25 
24 
23 
22 

11 
20^ 

19 
18 
17 
16 
15 
14 
13 
12 
11 


P.P. 


"  j  26 

25 

15 

SI    2.2 

2.1 

1.2 

10    4-3 

4.2 

2.S 

is'  6.5 

6.2 

3.8 

20i   8.7 

8.3 

S.o 

25,10.8 

10.4 

6.2 

30  13-0 

12.5 

7.S 

3S  IS.2 

14.0 

8.8 

40  17.3 

ib.7 

lO.O 

4S  19.5 

18.8 

II. 2 

S021.7 

20.8 

12.5 

ss:23.8 

22.9 

13.8 

14 

1.2 
2.3 
3-5 
4-7 
S.8 
7.0 
8.2 
9.3 
10.5 


5 
10 
IS 
20 
25 
30 
35 
40 
4S 
So;ii.7 
55  12.8 


11  I  10 

0.9  0.8 
1.8  1.7 

2.812.5 

3.7  3.3 

4.6|4.2 
5-55.0 
6.415.8 
7.3,6.7 
8.2  7.5 
9.28.3 
10.1I9.2 


P.P. 


41 


165 


, 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

*d. 

P.P. 

0 

9.81  694 

15 
14 

9.93  916 

26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 

0.06  084 

9.87  778 

11 
11 
11 
11 
11 
11 

11 
1 1 

60 

59 

1 

9.81  709 

9.93  942 

0.06  058 

9.87  767 

2 

9.81  723 

15 

14 

1 1; 

9.93  967 

0.06  033 

9.87  756 

58 

3 

9.81  738 

9.93  993 

0.06  007 

9.87  745 

57 

4 

9.81  752 

9.94  018 

0.05  982 

9.87  734 

56 

5 

9.81  767 

14 

15 
14 

9.94  044 

0.05  956 

9.87  723 

55 

6 

9.81  781 

9.94  069 

0.05  931 

9.87  712 

54 

7 

9.81  796 

9.94  095 

0.05  905 

9.87  701 

53 

8 

9.81  810 

15 
14 
15 
14 
14 

9.94  120 

0.05  880 

9.87  690 

11 
11 

11 
11 
11 

11 
11 
12 
11 
11 
11 

11 

11 
11 

52 

9 

9.81  825 

9.94  146 

0.05  854 

9.87  679 

51 

10 

9.81  839 

9.94  171 

0.05  829 

9.87  668 

50 

49 

11 

9.81  854 

9.94  197 

0.05  803 

9.87  657 

12 

9.81  868 

9.94  222 

0.05  778 

9.87  646 

48 

" 

26   25 

15 

13 

9.81  882 

15 

14 

9.94  248 

0.05  752 

9.87  635 

47 

5 

10 

2.2  2.1 

4.3  4  2 

1.2 
2  5 

14 

9.81  897 

9.94  273 

0.05  727 

9.87  624 

46 

IS 

6.5I  6.2 

3.8 

15 

9.81911 

15 

9.94  299 

0.05  701 

9.87  613 

45 

20:  8.71  8.3]  5.0 

2s;io.8  10.4  6.2 
30  13.0  i2.s:  7.5 

16 

9.81  926 

14 
15 
14 

9.94  324 

0.05  676 

9.87  601 

44 

17 

9.81  940 

9.94  350 

0.05  650 

9.87  590 

43 

35  is.2;i4.61  8.8 
40  17.3  16.7  lO.O 
45  19.5  18.8  II. 2 

18 

9.81  955 

9.94  375 

0.05  625 

9.87  579 

42 

19 

9.81  969 

14 
15 
14 

9.94  401 

0.05  599 

9.87  568 

41 
40 

SO  21.7  20.8  12.5 
5Si23.8l22.9ll3.8 

20 

9.81  983 

9.94  426 

0.05  574 

9.87  557 

21 

9.81  998 

9.94  452 

0.05  548 

9.87  546 

39 

22 

9.82  012 

14 

9.94  477 

26 
25 
26 
25 

0.05  523 

9.87  535 

11 

11 

12 

38 

23 

9.82  026 

15 
14 

9.94  503 

0.05  497 

9.87  524 

37 

24 

9.82  041 

9.94  528 

0.05  472 

9.87  513 

36 

25 

9.82  055 

14 

9.94  554 

0.05  446 

9.87  501 

11 

35 

26 

9.82  069 

15 
14 

9.94  579 

25 
26 
25 
26 
25 
26 
25 
26 
25 
26 

0.05  421 

9.87  490 

11 
11 
11 
11 
12 
11 
11 
11 
11 
12 
11 
11 
11 
11 
12 
11 
11 
12 
11 
11 
11 
12 
11 
11 
12 
11 
11 
12 
11 
11 
12 
11 
11 
12 

34 

27 

9.82  084 

9.94  604 

0.05  396 

9.87  479 

33 

28 

9.82  098 

14 

9.94  630 

0.05  370 

9.87  468 

32 

29 

9.82  112 

14 
15 
14 
14 
15 
14 

9.94  655 

0.05  345 

9.87  457 

31 

30 

9.82  126 

9.94  681 

0.05  319 

9.87  446 

30 

29 

31 

9.82  141 

9.94  706 

0.05  294 

9.87  434 

32 

9.82  155 

9.94  732 

0.05  268 

9.87  423 

28 

33 

9.82  169 

9.94  757 

0.05  243 

9.87  412 

27 

34 

9.82  184 

9.94  783 

0.05  217 

9.87  401 

26 

35 

9.82  198 

14 

9.94  808 

0.05  192 

9.87  390 

25 

36 

9.82  212 

14 
14 

9.94  834 

25 
9S 

0.05  166 

9.87  378 

24 

37 

9.82  226 

9.94  859 

0.05  141 

9.87  367 

23 

38 

9.82  240 

15 
14 
14 
14 

9.94  884 

26 

25 
26 

25 

0.05  116 

9.87  356 

22 

39 
40 

9.82  255 

9.94  910 

0.05  090 

9.87  345 

21 
20 

"     14  1  12  !  11 

9.82  269 

9  94  935 

0.05  065 

9.87  334 

41 

9.82  283 

9.94  961 

0.05  039 

9.87  322 

19 

5 

1.2  I.o'  0.0    1 

42 

9.82  297 

14 

9.94  986 

26 

25 
25 
26 
25 
26 
25 

0.05  014 

9.87  311 

18 

10 

2.3 

2.0J  1.8    1 

43 

9.82  311 

15 
14 

9.95  012 

0.04  988 

9.87  300 

17 

IS 
20 

3.5 

4.7 

3.0 

A   0 

2.8 
1  1 

44 

9.82  326 

9.95  037 

0.04  963 

9.87  288 

16 

25 

5.8 

5.0 

4.6 

45 

9.82  340 

14 

9.95  062 

0.04  938 

9.87  277 

15 

30 

7.0 
8.2 
Q.3 

6.0 

11 

■  7.3 

46 

9.82  354 

14 
14 

9.95  088 

0.04  912 

9.87  266 

14 

40 

8.0 

47 

9.82  368 

9.95  113 

0.04  887 

9.87  255 

13 

45  10.5 

9.0 

8.2 

48 

9.82  382 

14 

9.95  139 

0.04  861 

9.87  243 

12 

55  12.8 

II.O 

10. 1 

49 
50 

9.82  396 

14 
14 
15 

9.95  164 

26 

25 
25 
26 
25 
26 
25 

0.04  836 

9.87  232 

11 
10 

9 

9.82  410 

9.95  190 

0.04  810 

9.87  221 

51 

9.82  424 

9.95  215 

0.04  785 

9.87  209 

52 

9.82  439 

14 

9.95  240 

0.04  760 

9.87  198 

8 

53 

9.82  453 

14 
14 

9.95  266 

0.04  734 

9.87  187 

7 

54 

9.82  467 

9.95  291 

0.04  709 

9.87  175 

6 

55 

9.82  481 

14 

9.95  317 

0.04  683 

9.87  164 

5 

56 

9.82  495 

14 

14 

9.95  342 

26 

25 
7'^ 

0.04  658 

9.87  153 

4 

57 

9.82  509 

9.95  368 

0.04  632 

9.87  141 

3 

58 

9.82  523 

14 

9.95  393 

0.04  607 

9.87  130 

2 

59 

9.82  537 

14 

9.95  418 

26 

0.04  582 

9.87  119 

1 
0 

60 

9.82  551 

9.95  444 

0.04  556 

9.87  107 



L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

d. 

1 

P.P. 

166 

42° 

, 

L.  Sin. 

d. 

L.  Tan. 

c.d. 

L.  Cot. 

L.  Cos. 

d. 

P.P. 

0 

9.82  551 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
13 
14 
14 
13 
14 
13 
14 
14 
13 
14 
13 
14 
14 
13 
14 
13 

9.95  444 

25 
26 
25 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
26 
25 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 

0.04  556 

9.87  107 

11 
11 
12 

11 
12 
11 

11 
12 
11 
12 

11 
12 
11 
12 
11 
12 
11 
11 
12 

11 
12 
12 
11 
12 
11 
12 

11 
12 
11 
12 

11 
12 
12 
11 
12 
11 
12 
12 
11 
12 
12 
11 
12 
12 
11 
12 
12 
11 
12 

12 
12 
11 
12 
12 
11 
12 
12 
12 
11 
12 

60 

59 

58 
57 

1 

2 
3 

9.82  565 
9.82  579 
9.82  593 

9.95  469 
9.95  495 
9.95  520 

0.04  531 
0.04  505 
0.04  480 

9.87  096 
9.87  085 
9.87  073 

4 
5 
6 

9.82  607 
9.82  621 
9.82  635 

9.95  545 
9.95  571 
9.95  596 

0.04  455 
0.04  429 
0.04  404 

9.87  062 
9.87  050 
9.87  039 

56 

55 
54 

7 

8 

9 

10 

11 
12 
13 

9.82  649 
9.82  663 
9.82  677 

9.95  622 
9.95  647 
9.95  672 

0.04  378 
0.04  353 
0.04  328 

9.87  028' 
9.87  016 
9.87  005 

53 
52 
51 
50 
49 
48 
47 

9.82  691 

9.95  698 

0.04  302 

9.86  993 

9.82  705 
9.82  719 
9.82  733 

9.95  723 
9.95  748 
9.95  774 

0.04  277 
0.04  252 
0.04  226 

9.86  982 
9.86  970 
9.86  959 

"  1  26  25 

S'  2.2  2.1 

10  4-3  4-2 

14 

1.2 
2.3 

14 
15 
16 

9.82  747 
9.82  761 
9.82  775 

9.95  799 
9.95  825 
9.95  850 

0.04  201 
0.04  175 
0.04  150 

9.86  947 
9.86  936 
9.86  924 

46 

45 
44 

15  6.5  6.2  3-5 
20  8.7  8.3I  4.7 
25  10.8  10.4!  5.8 
30  13.0  12.51  7-0 

17 
18 
19 

9.82  788 
9.82  802 
9.82  816 

9.95  875 
9.95  901 
9.95  926 

0.04  125 
0.04  099 
0.04  074 

9.86  913 
9.86  902 
9.86  890 

43 
42 
41 
40 

39 

38 
37 

35  15-2  i4-6i  8.2 
40  17.3  16.7!  9.3 
45  19-5  18.8  10.5 
50  21.7  20.8  II.7 
55  23.8  22.9  12.8 

20 

9.82  830 

9.95  952 

0.04  048 

9.86  879 

21 
22 
23 

9.82  844 
9.82  858 
9.82  872 

9.95  977 

9.96  002 
9.96  028 

0.04  023 
0.03  998 
0.03  972 

9.86  867 
9.86  855 
9.86  844 

24 
25 
26 

9.82  885 
9.82  899 
9.82  913 

9.96  053 
9.96  078 
9.96  104 

0.03  947 
0.03  922 
0.03  896 

9.86  832 
9.86  821 
9.86  809 

36 

35 
34 

27 
28 
29 

9.82  927 
9.82  941 
9.82  955 

9.96  129 
9.96  155 
9.96  180 

0.03  871 
0.03  845 
0.03  820 

9.86  798 
9.86  786 
9.86  775 

33 
31 
31 
30 

29 

28 
27 

30 

9.82  968 

9.96  205 

0.03  795 

9.86  763 

31 
32 
33 

9.82  982 

9.82  996 

9.83  010 

9.96  231 
9.96  256 
9.96  281 

0.03  769 
0.03  744 
0.03  719 

9.86  752 
9.86  740 
9.86  728 

34 
35 
36 

9.83  023 
9.83  037 
9.83  051 

9.96  307 
9.96  332 
9.96  357 

0.03  693 
0.03  668 
0.03  643 

9.86  717 
9.86  705 
9.86  694 

26 
25 
24 

37 
38 
39 

9.83  065 
9.83  078 
9.83  092 

9.96  383 
9.96  408 
9.96  433 

0.03  617 
0.03  592 
0.03  567 

9.86  682 
9.86  670 
9.86  659 

23 
22 
21 
20 
19 
18 
17 

40 

9.83  106 

9.96  459 

0.03  541 

9.86  647 

"  13 

SI  i-i 
10  2.2 
15  3.2- 

12  ^^    II 

41 
42 
43 

9.83  120 
9.83  133 
9.83  147 

9.96  484 
9.96  510 
9.96  535 

0.03  516 
0.03  490 
0.03  465 

9.86  635 
9.86  624 
9.86  612 

I.O 

2.0 
3.0 

0.9 
1.8 
2.8 

44 
45 
46 

9.83  161 
9.83  174 
9.83  188 

9.96  560 
9.96  586 
9.96  611 

0.03  440 
0.03  414 
0.03  389 

9.86  600 
9.86  589 
9.86  577 

16 

15 
14 

20  4.3 
25  5.4 

30  6.5 

35  7.6 

4.0 
5-0 
6.0 
7.0 

80 

7-3 

47 
48 
49 

9.83  202 
9.83  215 
9.83  229 

9.96  636 
9.96  662 
9.96  687 

0.03  364 
0.03  338 
0.03  313 

9.86  565 
9.86  554 
9.86  542 

13 
12 
11 
10 

9 

8 

7 

45  9.8 
50  10.8 
55  I1.9 

9.0 

lO.O 
II.O 

8.2 
9.2 
lO.I 

50 

9.83  242 

9.96  712 

0.03  288 

9.86  530 

51 

52 
53 

9.83  256 
9.83  270 
9.83  283 

9.96  738 
9.96  763 
9.96  788 

0.03  262 
0.03  237 
0.03  212 

9.86  518 
9.86  507 
9.86  495 

54 
55 
56 

9.83  297 
9.83  310 
9.83  324 

9.96  814 
9.96  839 
9.96  864 

0.03  186 
0.03  161 
0.03  136 

9.86  483 
9.86  472 
9.86  460 

6 

5 
4 

57 
58 
59 

9.83  338 
9.83  351 
9.83  365 

9.96  890 
9.96  915 
9.96  940 

0.03  110 
0.03  085 
0.03  060 

9.86  448 
9.86  436 
9.86  425 

3 
2 
1 
0 

60 

9.83  378 

9.96  966 

0.03  034 

9.86  413 

L.  Cos. 

d. 

L.  Cot. 

c.d. 

L.  Tan. 

L.  Sin. 

d. 

' 

P.P. 

10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


L.  Sin. 


9.83  378 


9.83  392 
9.83  405 
9.83  419 
9.83  432 
9.83  446 
9.83  459 
9.83  473 
9.83  486 
9.83  500 


9.83  513 


9.83  527 
9.83  540 
9.83  554 
9.83  567 
9.83  581 
9.83  594 
9.83  608 
9.83  621 
9.83  634 


9.83  648 


9.83  661 
9.83  674 
9.83  688 
9.83  701 
9.83  715 
9.83  728 
9.83  741 
9.83  755 
9.83  768 


9.83  781 


9.83  795 
9.83  808 
9.83  821 
9.83  834 
9.83  848 
9.83  861 
9.83  874 
9.83  887 
9.83  901 


9.83  914 


9.83  927 
9.83  940 
9.83  954 
9.83  967 
9.83  980 

9.83  993 

9.84  006 
9.84  020 
9.84  033 


.84  046 


84  059 
84  072 
84  085 
84  098 
84  112 
84  125 
84  138 
84  151 
84  164 


84  177 


L.  Cos. 


14 

13 

14 

13 

14 

13 

14 

13 

14 

13 

14 

13 

14 

13 

14 

13 

14 

13 

13 

14 

13 

13 

14 

13 

14 

13 

13 

14 

13 

13 

14 

13 

13 

13 

14 

13 

13 

13 

14 

13 

13 

13 

14 

13 

13 

13 

13 

14 

13 

13 

13 

13 

13 

13 

14 

13 

13 

13 

13 

13 


43' 

L.  Tan.  c.d.  L.  Cot.   L.  Cos.   d. 


167 


9.96  966 


9.96  991 

9.97  016 
9.97  042 
9.97  067 
9.97  092 
9.97  118 
9.97  143 
9.97  168 
9.97  193 


9.97  219 


9.97  244 
9.97  269 
9.97  295 
9.97  320 
9.97  345 
9.97  371 
9.97  396 
9.97  421 
9.97  447 


9.97  472 


9.97  497 
9.97  523 
9.97  548 
9.97  573 
9.97  598 
9.97  624 
9.97  649 
9.97  674 
9.97  700 


9.97  725 


9.97  750 
9.97  776 
9.97  801 
9.97  826 
9.97  851 
9.97  877 
9.97  902 
9.97  927 
9.97  953 


9.97  978 


9.98  003 
9.98  029 
9.98  054 
9.98  079 
9.98  104 
9.98  130 
9.98  155 
9.98  180 
9.98  206 


9.98  231 


9.98  256 
9.98  281 
9.98  307 
9.98  332 
9.98  357 
9.98  383 
9.98  408 
9.98  433 
9.98  458 


9.98  484 


25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 

25 

25 

26 

25 

25 

25 

26 


0.03  034 


0.03  009 
0.02  984 
0.02  958 
0.02  933 
0.02  908 
0.02  882 
0.02  857 
0.02  832 
0.02  807 


0.02  781 


0.02  756 
0.02  731 
0.02  705 
0.02  680 
0.02  655 
0.02  629 
0.02  604 
0.02  579 
0.02  553 


0.02  528 


0.02  503 
0.02  477 
0.02  452 
0.02  427 
0.02  402 
0.02  376 
0.02  351 
0.02  326 
0.02  300 


0.02  275 


0.02  250 
0.02  224 
0.02  199 
0.02  174 
0.02  149 
0.02  123 
0.02  098 
0.02  073 
0.02  047 


0.02  022 


0.01  997 
0.01971 
0.01  946 
0.01921 
0.01  896 
0.01  870 
0.01  845 
0.01  820 
0.01  794 


9.86  413 


9.86  401 
9.86  389 
9.86  377 
9.86  366 
9.86  354 
9.86  342 
9.86  330 
9.86  318 
9.86  306 


9.86  295 
9.86  283 
9.86  271 
9.86  259 
9.86  247 
9.86  235 
9.86  223 
9.86  211 
9.86  200 
9.86  188 


9.86  176 


9.86  164 
9.86  152 
9.86  140 
9.86  128 
9.86  116 
9.86  104 
9.86  092 
9.86  080 
9.86  068 


9.86  056 


9.86  044 
9.86  032 
9.86  020 
9.86  008 
9.85  996 
9.85  984 
9.85  972 
9.85  960 
9.85  948 


9.85  936 


0.01  769 


0.01  744 
0.01  719 
0.01  693 
0.01  668 
0.01  643 
0.01  617 
0.01  592 
0.01  567 
0.01  542 


0.01  516 


9.85  924 
9.85  912 
9.85  900 
9.85  888 
9.85  876 
9.85  864 
9.85  851 
9.85  839 
9.85  827 


9.85  815 


9.85  803 
9.85  791 
9.85  779 
9.85  766 
9.85  754 
9.85  742 
9.85  730 
9.85  718 
9.85  706 


9.85  693 


L.  Cot.  c.d.  L.  Tan.   L.  Sin, 


^Ao 


30 


P.P. 


ir 

26 

25 

5 

2.2 

2.1 

lO 

4-3 

4.2 

IS 

6.5 

6.2 

20!  8.71  8.3 

25  10.8' 10.4 

30  13.0;  12. s 

35,15.2  14.6 

40 

17.316.7 

45 

19.5  18.8 

50 

21.7  20.8 

55 

23.8 

22.9I 

1.2 
2.3 
3.5 

4-7 
5.8 
7.0 
8.2 
9-3 
lo.s 
11.7 
12.8 


" 

13 

12 

11 

5  I.I 

I.O 

0.9 

10,  2.2 

2.0 

1.8 

15'  3-2 

3.0 

2.8 

20 

4.3 

4.0 

3.7 

25 

5.4 

5.0 

4.6 

30 

6.5 

6.0 

5.S 

35 

7.6 

7.0 

6.4 

40 

8.7 

8.0 

7.3 

45 

Q.8 

9.0 

8.2 

50 

10.8 

lO.O 

9.2 

55 

11.9 

II.O 

lO.I 

P.P. 


168 


44 


1 

2 
3 
4 
5 
6 
7 
8 

10 


30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


51 

52 
53 
54 
55 
56 
57 
58 

60 


L.  Sin.   d. 


9.84  177 


9.84  190 
9.84  203 
9.84  216 
9.84  229 
9.84  242 
9.84  255 
9.84  269 
9.84  282 
9.84  295 


9.84  308 


9.84  321 
9.84  334 
9.84  347 
9.84  360 
9.84  373 
9.84  385 
9.84  398 
9.84  411 
9.84  424 


9.84  437 


9.84  450 
9.84  463 
9.84  476 
9.84  489 
9.84  502 
9.84  515 
9.84  528 
9.84  540 
9.84  553 


9.84  566 


9.84  579 
9.84  592 
9.84  605 
9.84  618 
9.84  630 
9.84  643 
9.84  656 
9.84  669 
9.84  682 


9.84  694 


9.84  707 
9.84  720 
9.84  733 
9.84  745 
9.84  758 
9.84  771 
9.84  784 
9.84  796 
9.84  809 


9.84  822 


9.84  835 
9.84  847 
9.84  860 
9.84  873 
9.84  885 
9.84  898 
9.84  911 
9.84  923 
9.84  936 


9.84  949 


13 

13 

13 

13 

13 

13 

14 

13 

13 

13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

13 

13 

12 

13 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

13 

12 

13 

13 

12 

13 

13 

12 

13 

13 


L.  Tan. 


9.98  484 


9.98  509 
9.98  534 
9.98  560 
9.98  585 
9.98  610 
9.98  635 
9.98  661 
9.98  686 
9.98  711 


9.98  737 


9.98  762 
9.98  787 
9.98  812 
9.98  838 
9.98  863 
9.98  888 
9.98  913 
9.98  939 
9.98  964 


9.98  989 


9.99  015 
9.99  040 
9.99  065 
9.99  090 
9.99  116 
9.99  141 
9.99  166 
9.99  191 
9.99  217 


9.99  242 


9.99  267 
9.99  293 
9.99  318 
9.99  343 
9.99  368 
9.99  394 
9.99  419 
9.99  444 
9.99  469 


9.99  495 


9.99  520 
9.99  545 
9.99  570 
9.99  596 
9.99  621 
9.99  646 
9.99  672 
9.99  697 
9.99  722 


9.99  747 


9.99  773 
9.99  798 
9.99  823 
9.99  848 
9.99  874 
9.99  899 
9.99  924 
9.99  949 
9.99  975 


c.d. 


0.00  000 


25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 

26 

25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
26 
25 


L.  Cot. 


0.01  516 


0.01  491 
0.01  466 
0.01  440 
0.01  415 
0.01  390 
0.01  365 
0.01  339 
0.01  314 
0.01  289 


0.01  263 


L.  Cos.   d.   L.  Cot.  c.d.  L.  Tan.  L.  Sin 


0.01  238 
0.01  213 
0.01  188 
0.01  162 
0.01  137 
0.01  112 
0.01  087 
0.01  061 
0.01  036 
0.01011 
0.00  985 
0.00  960 
0.00  935 
0.00  910 
0.00  884 
0.00  859 
0.00  834 
0.00  809 
0.00  783 


L.  Cos.  I  d. 


9.85  693 


9.85  681 
9.85  669 
9.85  657 
9.85  645 
9.85  632 
9.85  620 
9.85  608 
9.85  596 
9.85  583 


9.85  571 


9.85  559 
9.85  547 
9.85  534 
9.85  522 
9.85  510 
9.85  497 
9.85  485 
9.85  473 
9.85  460 


0.00  758 


0.00  733 
0.00  707 
0.00  682 
0.00  657 
0.00  632 
0.00  606 
0.00  581 
0.00  556 
0.00  531 


0.00  505 


0.00  480 
0.00  455 
0.00  430 
0.00  404 
0.00  379 
0.00  354 
0.00  328 
0.00  303 
0.00  278 


0.00  253 


0.00  227 
0.00  202 
0.00  177 
0.00  152 
0.00  126 
0.00  101 
0.00  076 
0.00  051 
0.00  025 


0.00  000 


9.85  448 
9.85  436 
9.85  423 
9.85  411 
9.85  399 
9.85  386 
9.85  374 
9.85  361 
9.85  349 
9.85  337 


9.85  324 


9.85  312 
9.85  299 
9.85  287 
9.85  274 
9.85  262 
9.85  250 
9.85  237 
9.85  225 
9.85  212 


9.85  200 


9.85  187 
9.85  175 
9.85  162 
9.85  150 
9.85  137 
9.85  125 
9.85  112 
9.85  100 
9.85  087 


9.85  074 


9.85  062 
9.85  049 
9.85  037 
9.85  024 
9.85  012 
9.84  999 
9.84  986 
9.84  974 
9.84  961 


9.84  949 


P.P. 


10 


26  1  25 


2.2j    2.1 

4-3  4-2 
6.5  6.2 
8.7  8.3 
io.8jio.4 
13-0  12.5 
15.2  14.6 
17-3  16.7 
19.5J18.8 
SOJ2I.7;20.8 

SS:23.8!22.9 


14  ,  13      12 


1.2 

2.3 

3-5 

4-7 

5.8 

7.0 

8.2 

9-3 

10.5 

11.7 

12.8 


I.I 
2.2 

3-2 

4-3 
5-4 
6.5 
7.6 
8.7 
9.8 
10.8 
11.9 


i.o 
2.0 
30 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

lO.O 
II.O 


P.P. 


TABLE  III. 


NATURAL  TRIGONOMETRIC 
FUNCTIONS. 


169 


170        0° 


N.  Sin.  N.Tan.  N.  Cot.  N.  Cos. 


JO 

11 
12 
13 
14 
15 
16 
17 
18 

Jl 
20 


21 

22 
23 
24 
25 
26 
27 
28 

30 


31 
32 
33 
34 
35 
36 
37 
38 
39^ 
40 


41 

42 
43 
44 
45 
46 
47 
48 
49 


50 


51 

52 
53 
54 
55 
56 
57 
58 
59^ 
60 


.00000 


029 
058 
087 
116 
.00145 
175 
204 
233 
262 


.00291 


320 
349 
378 
407 
.00436 
465 
495 
524 
553 


.00582 


611 
640 
669 
698 
.00727 
756 
785 
814 
844 


.00873 


902 
931 
960 
.00989 
.01018 
047 
076 
105 
134 


.01164 


193 
222 
251 
280 
.01309 
33S 
367 
396 
425 


.01454 


483 
513 
542 
571 
.01600 
629 
658 
687 
716 


.01745 


.00000 


029 
058 
087 
116 
.00145 
175 
204 
233 
262 


.00291 


320 
349 
378 
407 
.00436 
465 
495 
524 
553 


.00582 


611 
640 
669 
698 
.00727 
756 
785 
815 
844 


,00873 


902 
931 
960 
,00989 
.01018 
047 
076 
105 
135 


.01164 


193 
222 
251 
280 
.01309 
338 
367 
396 
425 


.01455 


484 
513 
542 
571 
.01600 
629 
658 
687 
716 


.01746 


oo 


3437.7 
1718.9 
1145.9 
859.44 
687.55 
572.96 
491.11 
429.72 
381.97 


343.77 


312.52 
286.48 
264.44 
245.55 
229.18 
214.86 
202.22 
190.98 
180.93 


171.89 


163.70 
156.26 
149.47 
143.24 
137.51 
132.22 
127.32 
122.77 
118.54 


114.59 


110.89 
107.43 
104.17 
101.11 
98.218 
95.489 
92.908 
90.463 
88.144 


85.940 


83.844 
81.847 
79.943 
78.126 
76.390 
74.729 
73.139 
71.615 
70.153 


68.750 


67.402 
66.105 
64.858 
63.657 
62.499 
61.383 
60.306 
59.266 
58.261 


57.290 


N.  Cos.  N.  Cot.  N.Tan.  N.  Sin.  ' 


l.OCOO 


000 
000 
000 
000 
1.0000 
000 
000 
000 
000 


1.0000 


.99999 
999 
999 
999 

.99999 
999 
999 
999 
998 


.99998 


998 
998 
998 
998 
.99997 
997 
997 
997 
996 


.99996 


996 
996 
995 
995 
.99995 
995 
994 
994 
994 


.99993 


993 
993 
992 
992 
.99991 
991 
991 
990 
990 


.99989 


989 
989 
988 
988 
.99987 
987 
986 
986 
985 


,99985 


60 


59 

58  i 

57 

56 

55 

54 

53 

52 

51 


50 


49  I 

48 

47 

46 

45 

44 

43 

42 

41 


40 


39 
38 
37 
36 
35 
34 
33 
31 

IL 
30 


19  j 

18  I 
17  \ 
16  ! 
15  i 
14  ' 
13  ' 
12 
11  I 


10 

7  I 
6  ! 
5  I 
4  j 
3  ! 
2 
J_ 
0 


1 

o 

1  ' 
1 

N.  Sin. 

N.Tan.! N.  Cot. 

N.  Cos. 

[» 

.01745 

.01746  157.290 

.99985 

60 

59 

774 

775 

56.351 

984 

1  2 

803 

804 

55.442 

«>S4 

58 

1  3 

832 

833 

54.561 

933 

57 

1  4 

862 

862 

53.709 

983 

56 

5 

.01891 

.01891 

52.882 

.999«2 

55 

i  ^ 

920 

920 

52.081 

982 

54 

7 

949 

949 

51.303 

981 

53 

8 

.01978 

.01978 

50.549 

980 

52 

:  9 
i  10 

.02007 

.02007 

49.816 

980 

51 

.02036 

.02036 

4t).104 

.99979 

50 

i  11 

065 

066 

48.412 

979 

49 

'  12 

094 

095 

47.740 

978 

48 

;  13 

123 

124 

47.085 

977 

47 

Il4 

152 

153 

46.449 

977 

46 

15 

.02181 

.02182 

45.829 

.99976 

45 

16 

211 

211 

45.226 

976 

44 

i  17 

240 

240 

44.639 

975 

43 

i  18 

269 

269 

44.066 

974 

42 

;  19 
20 

298 

298 

43.508 

974 

41 
40 

39 

.02327 

.02328 

42.964 

.99973 

21 

356 

357 

42.433 

972 

22 

385 

386 

41.916 

972 

38 

:  23 

414 

415 

41.411 

971 

37 

!  24 

443 

444 

40.917 

970 

36 

1  25 

.02472 

.02473 

40.436 

.99969 

35 

26 

501 

502 

39.965 

969 

34 

27 

530 

531 

39.506 

968 

33 

28 

560 

560 

39.057 

967 

32 

29 
30 

589 

589 

38.618 

966 

31 
30 

.02618 

.02619 

38.188 

.99966 

31 

647 

648 

37.769 

965 

29 

32 

676 

677 

37.358 

964 

28 

33 

705 

706 

36.956 

963 

27 

\  34 

734 

735 

36.563 

963 

26 

■  35 

.02763 

.02764 

36.178 

.99962 

25 

36 

792 

793 

35.801 

961 

24 

37 

821 

822 

35.431 

960 

23 

3% 

850 

851 

35.070 

959 

22 

39 
40 

879 

881 

34.715 

959 

21 
20 

19 

.02908 

.02910 

34.368 

.99958 

41 

938 

939 

34.027 

957 

■  42 

967 

968 

33.694 

956 

18 

43 

.02996 

.02997 

33.366 

955 

17 

\  44 

.03025 

.03026 

33.045 

954 

16 

:  45 

.03054 

.03055 

32.730 

.99953 

15 

1  46 

083 

084 

32.421 

952 

14 

i  47 

112 

114 

32.118 

952 

13 

'  48 

141 

143 

31-821 

951 

12 

:  49 
50 

170 

172 

31.528 

950 

11 
10| 

9 

.03199 

.03201 

31.242 

.99949 

51 

228 

230 

30.960 

948 

^  52 

257 

259 

30.683 

947 

8 

1  53 

286 

288 

30.412 

946 

7 

i  54 

316 

317 

30.145 

945 

6 

!  55 

.03345 

.03346 

29.882 

.99944 

5 

i  56 

374 

376 

29.624 

943 

4 

57 

403 

405 

29.371 

942 

3 

58 

432 

434 

29.122 

941 

2 

59 
60 

461 

463 

28.877 

940 

1 
0 

.03490 

.03492 

28.636 

.99939 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

'  i 

1 

2 

o 

1 

N.  Sm.|N.Tan.'N.  Cot.N.Cos 

!  0 

[ 

3 

.03490 

.03492 

28.636 

.99939 

60 

519 

548 
577 

521 
550 
579 

.399 
28.166 
27.937 

938 
937 
936 

59 

58 
57 

4 
5 
6 

606 

.03635 

664 

609 

.03638 

667 

.712 

27.490 

.271 

935 

.99934 

933 

56 

55 
54 

7 

8 

9 

10 

693 
723 

752 

696 

725 
754 

27.057 

26.845 

.637 

932 
931 
930 

53 
52 
51 
50 

.03781 

.03783 

26.432 

.99929 

11 
12 
13 

810 
839 
868 

812 
842 
871 

.230 
26.031 
25.835 

927 
926 
925 

49 
48 
47 

14 
15 

1  16 

897 

.03926 

955 

900 

.03929 

958 

.642 

25.452 

.264 

924 

.99923 

922 

46 

44  I 

17 

18 
19 

120 

.03984 

.04013 

042 

.03987 

.04016 

046 

25.080 

24.898 

.719 

921 
919 
918 

43 
42 
41 
40 

.04071 

.04075 

24.542 

.99917 

21 
22 
23 

100 
129 
159 

104 
133 
162 

.368 

.196 

24.026 

916 
915 
913 

39 
38 
37 

24 
25 
26 

188 

.04217 

246 

191 

.04220 

250 

23.859 

23.695 

.532 

912 

.99911 

910 

36   1 

35 

34 

27 

28 

29 

,30 

275 
304 
333 

279 

308 
337 

.372 

.214 

23.058 

909 
907 
906 

33 
31 
31  1 
30 

.04362 

.04366 

22.904 

.99905 

31 
32 
33 

391 
420 
449 

395 
424 
454 

.752 
.602 
.454 

904 
902 
901 

29  , 

28 

27  1 

34 

i  35 
i  36 

478 

.04507 

536 

483 

.04512 

541 

.308 
22.164 
22.022 

900 

99898 

897 

26  1 

37 
38 
39 

565 
594 
623 

570 
599 
628 

21.881 
.743 
.606 

896 
894 
893 

13 
11 
21 

40 

.04653 

.04658 

21.470 

.99892 

20 

41 
42 
43 

682 
711 
740 

687 
716 
745 

.337 

.205 

21.075 

890 
889 
888 

19 

18  ; 

17 

44 
45 
46 

769 
.04798 

827 

774 

.04803 

833 

20.946 

20.819 

.693 

886 

.99885 

883 

16 
15 
14 

47 
48 
49 
50 

856 
885 
914 

862 
891 
920 

.569 
.446 
.325 

882 
881 
879 

13 
12 
11 
10 

.04943 

.04949 

20.206 

.99878 

51 
52 
53 

.04972 

.05001 

030 

.04978 

.05007 

037 

20.087 
19.970 

.855 

876 
875 
873 

9 

8 
7 

54 
55 
56 

059 

.05088 

117 

066 

.05095 

124 

.740 

19.627 

.516 

872 

.99870 

869 

6 

5  ! 

4 

57 
58 
59 
60 

146 

175 
205 

153 
182 
212 

.405 
.296 
.188 

867 
866 
864 

3 

2  \ 

1  i 

.05234 

.05241 

19.081 

.99863 

|n.  Cos. 

N.  Cot. 

N.Tan.|N.  Sin.| 

'  1 

3° 

171 

1  , 

0 

N.  Sin 

'N.Tan.  N.  Cot 

^N.Cos 

.05234 

.05241 

19.081 

.99863 

60 

59 

58 
57 

1 

2 

i  3 

263 
292 
321 

270 
299 
328 

18.976 
.871 
.768 

861 
860 

858 

1  4 
5 
6 

350 

.05379 

408 

357 

.05387 

416 

.666 

18.564 

.464 

857 

.99855 

854 

56 
55 
54 

7 

i  8 

9 

437 
466 
495 

445 
474 
503 

.366 
.268 
.171 

852 
851 
849 

53 
52 
51 

10 

.05524 

.05533 

18.075 

.99847 

50 

11 
12 
13 

553 
582 
611 

562 
591 
620 

17.980 
.886 
.793 

846 
844 
842 

49 

48 
47 

14 
15 
16 

640 

.05669 

698 

649 

.05678 

708 

.702 

17.611 

.521 

841 

.99839 

838 

46 
45 

44 

17 
18 
19 
20 

727 
756 
785 

737 
766 
.795 

.431 
.343 
.256 

836 
834 
833 

43 
42 
41 
40 

.05814 

.05824 

17.169 

.99831 

21 
22 
23 

844 
873 
902 

854 
883 
912 

17.084 

16.999 

.915 

829 
827 
826 

39 
38 
37 

1  24 
25 
26 

931 

.05960 

989 

941 

.05970 

999 

.832 

16.750 

.668 

824 

.99822 

821 

36 
31 
34 

27 
28 
29 
30 

.06018 
047 
076 

.06029 

058 
087 

.587 
507 
.428 

819 

817 
815 

33 
31 
31 
30 

.06105 

.06116 

16.350 

.99813 

31 
32 
33 

134 
163 
192 

145 
175 
204 

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812 
810 
808 

29 

28 
27 

34 

35 

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221 

.06250 

279 

233 

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291 

16.043 

15.969 

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806 

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803 

26 
25 
24 

1  37 

38 

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308 
337 
366 

321 
350 
379 

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801 
799 
797 

23 
22 
21 

40 

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15.605 

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20 

41 
42 
43 

424 
453 
482 

438 
467 
496 

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793 
792 
790 

19 
18 
17 

44 
45 
46 

511 

.06540 

569 

525 
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584 

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15.257 

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788 

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784 

16 
15 
14 

47 
48 
49 
50 

598 
627 
656 

613 
642 
671 

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15.056 
14.990 

782 
780 
778 

13 
12 
11 

.06685 

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14.924 

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10 

51 
52 
53 

714 
743 
773 

730 
759 

788 

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774 
772 
770 

9 

8 

7 

54 
55 
56 

802 

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860 

817 

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876 

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14.606 

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768 

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6 

5 
4 

57 
58 
59 
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918 
947 

905 
934 
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762 
760 

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14.301 

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N.  Sin. 

/ 

172 

4 

o 

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N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.06976 

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14.301 

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60 

1  1 
2 
3 

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063 

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5 
6 

092 

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150 

110 

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168 

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14.008 
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56 
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7 
8 
9 

179 

208 
237 

197 

227 
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740 
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53 
52 
51 

10 

11 
12 
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295 
324 
353 

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314 
344 
373 

13.727 

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48 
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14 
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13.457 

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45 

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17 
18 
19 
20 

21 
22 
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527 

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710 
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43 
42 
41 
40 
39 
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614 
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13.197 

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13.046 

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25 
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12.996 

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36 

35 
34 

27 
28 
29 
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817 

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812 

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33 
32 
31 
30 

29 
28 
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689 
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12.474 

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41 
42 
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165 
194 

223 

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221 
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18 
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44 
45 
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310 

280 

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47 
48 
49 
50 

51 
52 
53 

339 

368 

397 

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368 

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625 

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N.Tan.  N.  Cot. 

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" 

0 

.08716 

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11.430 

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60 

1 

2 

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11 

12 
13 

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101 
130 

11.024 

10.988 

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588 
586 

49 
48 

47 

14 

15 

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179 

159 

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45 
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19 

208 
237 
266 

247 
277 
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42 
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40 

39 

38 
37 

21 
22 
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28 

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31 
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537 
534 
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29 

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34 
35 
36 

700 
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10.229 

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528 
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26 
25 
24 

37 
38 
39 
40 

41 

42 
43 

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517 
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23 
22 
21 
20 
19 
18 
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10.078 

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506 
503 

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9.9893 

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45 
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040 

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9.9310 

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16 
15 
14 

47 
48 
49 
50 

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106 
135 

128 
158 
187 
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246 
275 
305 

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9.7882 
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488 
485 

13 
12 
11 
10 

9 

8 

7 

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479 
476 
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51 

52 
53 

192 
221 

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54 
55 
56 

279 

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337 

334 

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9.6493 

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58 
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458 
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9.5144 

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N.  Cos.N.  Cot. 

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6 

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0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

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1 

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599 
628 

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9.5144 

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446 
443 
440 

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434 
431 
428 
424 

60  1 

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56  ' 

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50 

49 

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9.3831 
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9.2553 

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11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

771 
800 
829 
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234 
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418 
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412 
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383 
380 
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370 
367 
364 
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39 

38   ; 

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34  I 

33  i 

32  i 

31  ' 

30 

29 

28 

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25 

24 

23 

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21 

20 

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354 
351 
347 
344 

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334 
331 
327 

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8.2434 
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320 
317 
314 
310 
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303 
300 
297 
293 

19 
18 
17 
16 
15 
14 
13 
12 
11 
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8 
7 
6 
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N.  Sin. 

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7 

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173 

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0 

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1 

216 

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245 

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6 

360 

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7 

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7.9530 

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533 

633 

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13 

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18 

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/ 

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82° 


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8 

° 

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N.  Sin. 

N.  Tan. 

N.  Cot. 

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0 

1 

2 
3 

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55 
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10 

11 
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18 
19 
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38 
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21 
22 
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35 
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18 
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44 
45 
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184 

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241 

362 

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421 

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6.4971 

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16 
15 
14 

47 
48 
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270 

299 

327 

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6.4348 

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51 

52 
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570 
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54 
55 
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6.3737 

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N.  Cos. 

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9° 

1 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

60 

59 

58 
57 

0 

1 

2 
3 

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6.3138 

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11 
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6.1970 

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53 
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14 
15 
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6.1402 

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46 
45 
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17 
18 
19 
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21 
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"6:0844 
.0734 
.0624 
.0514 

690 
686 
681 

43 
42 
41 

.16218 
246 
275 
304 

.98676 

40 

465 
495 

525 

671 
667 
662 

39 

38 
37 

24 
25 
26 

333 

.16361 

390 

555 

.16585 

615 

.0405 

6.0296 

.0188 

657 

.98652 

648 

36 
35 
34 

27 
28 
29 
30 

419 

447 

476 

.16505 

645 
674 
704 
.16734 
764 
794 
824 

6.0080 

5.9972 

.9865 

643 

638 

633 

.98629 

33 
31 
31 
30 

5.9758 
.9651 
.9545 
.9439 

31 
32 
33 

533 
562 
591 

624 
619 
614 

29 

28 
27 

34 
35 
36 

620 

.16648 

677 

854 

.16884 

914 

.9333 

5.9228 
.9124 

609 

.98604 

600 

26 

25 
24 

37 
3^ 
39 
40 

41 
42 
43 

706 
734 
763 

944 
.16974 
.17004 

.9019 
.8915 
.8811 
5.8708 
.8605 
.8502 
.8400 

595 
590 

585 

23 
22 
21 
20 
19 
18 
17 

.16792 

.17033 
063 
093 
123 

.98580 

820 
849 
878 

575 
570 
565 

44 
45 
46 

906 

.16935 

964 

153 

.17183 

213 

.8298 

5.8197 

.8095 

561 

.98556 

551 

16 
15 
14 

47 
48 
1  49 
50 
51 
52 
53 

.16992 

.17021 

050 

243 
273 
303 

.7994 
.7894 
.7794 

546 
541 
536 

13 
12 
11 
10 

.17078 

.17333 

5.7694 

.98531 

107 
136 
164 

363 
393 

423 

.7594 
.7495 
.7396 

526 
521 
516 

9 

8 

7 

54 
55 
56 

193 

.17222 
250 

453 

.17483 

513 

.7297 

5.7199 

.7101 

511 

.98506 

501 

6 

5 
4 

57 
58 
59 

279 
308 
336 

543 
573 
603 

.7004 
.6906 
.6809 

496 
491 
486 

3 
2 
1 
0 
/ 

60 

.17365 

.17633 

5.6713 

.98481 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

^1° 


<so° 


10° 

11° 

175 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

! 

1  , 
! 

i  ^ 

N.  Sin. 

N.Tan.  N.  Cot. 

N.  Cos. 

1  0 

.17365 

.17633 

5.6713 

.98481 

60 

.19081 

.19438 

5.1446 

.98163 

60 

1 

393 

663 

.6617 

476 

59 

1  1 

109 

468 

.1366 

157 

59 

2 

422 

693 

.6521 

471 

58 

1  2 

138 

498 

.1286 

152 

58 

3 

451 

723 

.6425 

466 

57 

!  3 

167 

529 

.1207 

146 

57 

4 

479 

753 

.6329 

461 

56 

4 

195 

559 

.1128 

140 

56 

5 

^7508 

.17783 

5.6234 

.98455 

55 

i  5 

.19224 

.19589 

5.1049 

.98135 

55 

6 

537 

813 

.6140 

450 

54 

!  6 

252 

619 

.0970 

129 

54 

7 

565 

843 

.6045 

445 

53 

I  7 

281 

649 

.0892 

124 

53 

8 

594 

873 

.5951 

440 

52 

8 

309 

680 

.0814 

118 

52 

9 

623 

903 

.5857 

435 

51 
50 

49 

9 
10 

338 
.19366 

710 

.0736 

112 

51 
50 

49 

10 

11 

.17651 

.17933 

5.5764 

.98430 

.19740 

5.0658 

.98107 

680 

963 

.3671 

425 

11 

395 

770 

.0581 

101 

12 

708 

.17993 

.5578 

420 

48 

12 

423 

801 

.0504 

096 

48 

13 

737 

.18023 

.5485 

414 

47 

1  13 

452 

831 

.0427 

090 

47 

14 

766 

053 

.5393 

409 

46 

14 

481 

861 

.0350 

084 

46 

15 

.17794 

.18083 

5.5301 

.98404 

45 

15 

.19509 

.19891 

5.0273 

.98079 

45 

16 

823 

113 

.5209 

399 

44 

16 

538 

921 

.0197 

073 

44 

17 

852 

143 

.5118 

394 

43 

17 

566 

952 

.0121 

067 

43 

18 

880 

173 

.5026 

389 

42 

18 

595 

.19982 

5.0045 

061 

42 

19 

909 

203 

.4936 

383 

41 

19 

623 

.20012 

4.9969 

056 

41 
40 

20 

.17937 

.18233 

5.4845 

.98378 

40 

20 

21 

.19652 

.20042 

4.9894 

.98050 

21 

966 

263 

.4755 

373 

39 

680 

073 

.9819 

044 

39 

22 

.17995 

293 

.4665 

368 

38 

22 

709 

103 

.9744 

039 

38 

23 

.18023 

323 

.4575 

362 

37 

23 

737 

.133 

.9669 

033 

37 

24 

052 

353 

.4486 

357 

36 

24 

766 

164 

.9594 

027 

36 

25 

.18081 

.18384 

5.4397 

.98352 

35 

25 

.19794 

.20194 

4.9520 

.98021 

35 

26 

109 

414 

.4308 

347 

34 

26 

823 

224 

.9446 

016 

34 

27 

138 

444 

.4219 

341 

33 

27 

851 

254 

.9372 

010 

33 

28 

166 

474 

.4131 

336 

31 

28 

880 

285 

.9298 

.98004 

31 

29 

195 

504 

.4043 

331 

31 

29 
30 

908 

315 

.9225 

.97998 
.97992 

31 
30 

29 

30 

31 

.18224 

.18534 

5.3955 

.98325 

30 

29 

.19937 

.20345 

4.9152 

252 

564 

.3868 

320 

31 

965 

376 

.9078 

987 

32 

281 

594 

.3781 

315 

28 

32 

.19994 

406 

.9006 

981 

28 

33 

309 

624 

.3694 

310 

27 

33 

.20022 

436 

.8933 

975 

27 

34 

338 

654 

.3607 

304 

26 

34 

051 

466 

.8860 

969 

26 

i  35 

.18367 

.18684 

5.3521 

.98299 

25 

35 

.20079 

.20497 

4.8788 

.97963 

25- 

36 

395 

714 

.3435 

294 

24 

36 

108 

527 

.8716 

958 

24 

37 

424 

745 

.3349 

288 

23 

37 

136 

557 

.8644 

952 

23 

38 

452 

775 

.3263 

283 

22 

i  38 

165 

588 

.8573 

946 

22 

39 

481 

805 

.3178 

277 

21 
20 

19 

i39 
i  40 

193 

618 

.8501 

940 

21 

40 

41 

.18509 

.18835 

5.3093 

.98272 

.20222 

.20648 

4.8430 

.97934 

20 

19 

538 

865 

.3008 

267 

41 

250 

679 

.8359 

928 

42 

567 

895 

.2924 

261 

18 

|42 

279 

709 

.8288 

922 

18 

43 

595 

925 

.2839 

256 

17 

43 

307 

739 

.8218 

916 

17 

44 

624 

955 

.2755 

250 

16 

I44 

336 

770 

.8147 

910 

16 

45 

.18652 

.18986 

5.2672 

.98245 

15 

I  45 

.20364 

.20800 

4.8077 

.97905 

15 

46 

681 

.19016 

.2588 

240 

14 

46 

393 

830 

.8007 

899 

14 

47 

710 

046 

.2505 

234 

13 

47 

421 

861 

.7937 

893 

13 

48 

738 

076 

.2422 

229 

12 

i  48 

450 

891 

.7867 

887 

12 

49 

767 

106 

.2339 

223 

11 

1  49 
|50 

478 

921 

.7798 

881 

11 

50 

.18795 

.19136 

5.2257 

.98218 

10 

.20507 

.20952 

4.7729 

.97875 

10 

51 

824 

166 

.2174 

212 

9 

1  51 

535 

.20982 

.7659 

869 

9 

52 

852 

197 

.2092 

207 

8 

52 

563 

.21013 

.7591 

863 

8 

53 

881 

227 

.2011 

201 

7 

I  53 

592 

043 

.7522 

857 

7 

54 

910 

257 

.1929 

196 

6 

54 

620 

073 

.7453 

851 

6 

55 

.18938 

.19287 

5.1848 

.98190 

5 

55 

.20649 

.21104 

4.7385 

.97845 

5 

56 

967 

317 

.1767 

185 

4 

56 

677 

134 

.7317 

839 

4 

57 

.18995 

347 

.1686 

179 

3 

I57 

706 

164 

.7249 

833 

3 

58 

.19024 

378 

.1606 

174 

2 

1  58 

734 

195 

.7181 

827 

2 

59 

052 

408 

.1526 

168 

1 

i59 
!  60 

763 

225 

.7114 

821 

1 
0 

60 

.19081 

.19438 

5.1446 

.98163 

0 

.20791 

.21256 

4.7046 

.97815 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

i 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

79' 


78° 


176 

12° 

r^" 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

1 

2 
3 

.20791 

.21256 

4.7046 

.97815 

60 

820 
848 

877 

286 
316 

347 

.6979 
.6912 

.6845 

809 
803 
797 

59 

58 
57 

4 
5 
6 

905 

.20933 

962 

377 

.21408 

438 

.6779 

4.6712 

.6646 

791 
.97784 

778 

56 

55 
54 

7 
8 
9 

.20990 

.21019 

047 

469 
499 
529 

.6580 
.6514 
.6448 

772 
766 
760 

53 
52 
51 

10 

.21076 

.21560 

4.6382 

.97754 

50 

11 
12 
13 

104 
132 
161 

590 
621 
651 

.6317 
.6252 
.6187 

748 
742 
735 

49 
48 
47 

14 
15 
16 

189 

.21218 

246 

682 

.21712 

743 

.6122 

4.6057 

.5993 

729 
.97723 

717 

46 

45 
44 

17 
18 
19 
20 

275 
303 
331 

773 
804 
834 

.5928 
.5864 
.5800 

711 
705 
698 

43 
42 
41 
40 

.21360 

.21864 

4.5736 

.97692 

21 
22 
23 

388 
417 
445 

895 
925 
956 

.5673 
.5609 
.5546 

686 
680 
673 

39 

38 
37 

24 
25 
26 

474 

.21502 

530 

.21986 

.22017 

047 

.5483 

4.5420 

.5357 

667 

.97661 

655 

36 
35 
34 

27 
28 
29 
30 
31 
32 
33 

559 
587 
616 

078 
108 
139 

.5294 
.5232 
.5169 

648 
642 
636 

33 
31 
31 
30 

.21644 

.22169 
200 
231 
261 

4.5107 

.97630 

672 
701 
729 

.5045 
.4983 
.4922 

623 
617 
611 

29 

28 

27 

34 
35 
36 

758 

.21786 

814 

292 

.22322 
353 

.4860 

4.4799 

.4737 

604 

.97598 

592 

26 

25 
24 

37 
38 
39 

843 
871 
899 

383 
414 
444 

.4676 
.4615 

.4555 

585 
579 
573 

23 
22 
21 
20 

19 

18 
17 

40 

.21928 

.22475 

4.4494 

.97566 

41 
42 

43 

956 
.21985 
.22013 

505 
536 
567 

.4434 
.4373 
.4313 

560 

553 

547 

44 
45 
46 

041 

.22070 

098 

597 

.22628 

658 

.4253 

4.4194 

.4134 

541 
.97534 

528 

16 
15 
14 

47 
48 
49 

126 
155 
183 

689 
719 
750 

.4075 
.4015 
.3956 

521 

515 

508 

.97502 

13 
12 
11 
10 

50 

.22212 

.22781 

4.3897 

51 

52 
53 

240 
268 
297 

811 
842 
872 

.3838 
.3779 
.3721 

496 
489 

483 

9 

8 

7 

54 
55 
56 

325 

.22353 

382 

903 

.22934 
964 

.3662 

4.3604 

.3546 

476 

.97470 

463 

6 

5 
4 

57 
58 
59 

410 

438 
467 

.22995 

.23026 

056 

.3488 
.3430 
.3372 

457 
450 
444 

3 
2 
1 
0 

60 

.22495 

.23087 

4.3315 

.97437 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

13' 


1 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

1 

2 
3 

.22495 

.23087 

4.3315 

.97437 

60 

59 

58 
57 

523 
552 
580 

117 
148 
179 

.3257 
.3200 
.3143 

430 
424 
417 

4 
5 
6 

608 

.22637 

665 

209 

.23240 

271 

.3086 

4.3029 

.2972 

411 

.97404 

398 

56 

55 
54 

7 
8 
9 

693 
722 
750 

301 
332 
363 

.2916 
.2859 
.2803 

391 

384 
378 

53 
52 
51 
50 

10 

.22778 

.23393 

4.2747 

.97371 

11 
12 
13 

807 
835 
863 

424 

455 
485 

.2691 
.2635 
.2580 

365 
358 
351 

49 

48 

47 

14 
15 
16 

892 

.22920 

948 

516 

.23547 
578 

.2524 

4.2468 

.2413 

345 

.97338 

331 

46 

45 
44 

17 
18 
19 
20 

.22977 

.23005 

033 

608 
639 
670 

.2358 
.2303 
.2248 

325 
318 
311 

43 
42 
41 

.23062 

.23700 

4.2193 

.97304 

40 

21 
22 
23 

090 
118 
146 

731 
762 
793 

.2139 
.2084 
.2030 

298 
291 

284 

39 

38 
37 

24 
25 
26 

175 

.23203 

231 

823 

.23854 

885 

.1976 

4.1922 

.1868 

278 

.97271 

264 

36 

35 
34 

27 
28 
29 

260 
288 
316 

916 

946 

.23977 

.1814 
.1760 
.1706 

257 
251 
244 

33 
32 
31 

30 

.23345 

.24008 

4.1653 

.97237 

30 

31 

32 
33 

373 
401 
429 

039 
069 
100 

.1600 
.1547 
.1493 

230 
223 
217 

29 

28 
27 

34 
35 
36 

458 

.23486 

514 

131 

.24162 

193 

.1441 

4.1388 
.1335 

210 

.97203 

196 

26 

25 
24 

37 
38 
39 
40 

41 

42 
43 

542 
571 
599 

223 
254 
285 

.1282 
.1230 
.1178 

189 
182 
176 

23 
22 
21 
20 
19 
18 
17 

.23627 

.24316 

4.1126 

.97169 

656 
684 
712 

347 
377 
408 

.1074 
.1022 
.0970 

162 
155 

148 

44 
45 
46 

740 

.23769 

797 

439 

.24470 

501 

.0918 

4.0867 

.0815 

141 

.97134 

127 

16 

15 
14 

47 
48 
49 
50 

825 
853 
882 

532 
562 
593 

.0764 
.0713 
.0662 

120 
113 
106 

13 
12 
11 
10 

.23910 

.24624 

4.0611 

.97100 

51 

52 
53 

938 

966 

.23995 

655 
686 
717 

.0560 
.0509 
.0459 

093 
086 
079 

9 

8 

7 

54 
55 
56 

.24023 

.24051 

079 

747 

.24778 

809 

.0408 

4.0358 

.0308 

072 

.97065 

058 

6 

5 
4 

57 
58 
59 

108 
136 
164 

840 
871 
902 

.0257 
.0207 
.0158 

051 
044 
037 

3 
2 
1 

60 

.24192 

.24933 

4.0108 

.97030 

0 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

77° 


76° 


14° 


' 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.24192 

.24933 

4.0108 

.97030 

60 

1 

220 

964 

.0058 

023 

59 

2 

249 

.24995 

4.0009 

015 

58 

3 

277 

.25026 

3.9959 

008 

57 

4 

305 

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.9910 

.97001 

56 

5 

.24333 

.25087 

3.9861 

.96994 

55 

6 

362 

118 

.9812 

987 

54 

7 

390 

149 

.9763 

980 

53 

8 

418 

180 

.9714 

973 

52 

9 

446 

211 

.9665 

966 

51  1 
50  1 

10 

.24474 

,25242 

3.9617 

.96959 

11 

503 

273 

.9568 

952 

49 

12 

531 

304 

.9520 

945 

48 

13 

559 

335 

.9471 

937 

47 

14 

587 

366 

.9423 

930 

46 

15 

.24615 

.25397 

3.9375 

.96923 

45  1 

16 

644 

428 

.9327 

916 

44 

17 

672 

459 

.9279 

909 

43 

18 

700 

490 

.9232 

902 

42 

19 

728 

521 

.9184 

894 

41 

20 

.24756 

.25552 

3.9136 

.96887 

40 

21 

784 

583 

.9089 

880 

39 

22 

813 

614 

.9042 

873 

38 

23 

841 

645 

.8995 

866 

37 

24 

869 

676 

.8947 

858 

36 

25 

.24897 

.25707 

3.8900 

.96851 

35 

26 

925 

738 

.8854 

844 

34 

27 

954 

769 

.8807 

837 

33 

28 

.24982 

800 

.8760 

829 

31 

!  29 
30 

.25010 

831 

.8714 

822 

31 

.25038 

.25862 

3.8667 

.96815 

30 

31 

066 

893 

.8621 

807 

29 

32 

094 

924 

.8575 

800 

28 

33 

122 

955 

.8528 

793 

27 

34 

151 

.25986 

.8482 

786 

26 

35 

.25179 

.26017 

3.8436 

.96778 

25 

36 

207 

048 

.8391 

771 

24 

37 

235 

079 

.8345 

764 

23 

38 

263 

110 

.8299 

756 

22 

39 
40 

291 

141 

.8254 

749 

21 

.25320 

.26172 

3.8208 

.96742 

20 

41 

348 

203 

.8163 

734 

19 

42 

376 

235 

.8118 

727 

18 

43 

404 

266 

.8073 

719 

17  1 

44 

432 

297 

.8028 

712 

16 

45 

.25460 

.26328 

3.7983 

.96705 

15 

46 

488 

359 

.7938 

697 

14 

47 

516 

390 

.7893 

690 

13 

48 

545 

421 

.7848 

682 

12  ! 

49 

573 

452 

.7804 

675 

11 

1  50 

.25601 

.26483 

3.7760 

.96667 

10 

51 

629 

515 

.7715 

660 

9  i 

52 

657 

546 

.7671 

653 

8  1 

53 

685 

577 

.7627 

645 

7 

54 

713 

608 

.7583 

638 

6 

55 

.25741 

.26639 

3.7539 

.96630 

5 

56 

769 

670 

.7495 

623 

4 

57 

798 

701 

.7451 

615 

^  1 

58 

826 

733 

.7408 

608 

2  ! 

59 
60 

854 

764 

.7364 

600 
.96593 

1  1 
0  i 

.25882 

.26795 

3.7321 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

15° 

177 

\_ 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

.25882 

.26795 

3.7321 

.96593 

60 

59 

58 
57 
56 
55 
54 
53 
52 
51 

910 
938 
966 
.25994 
.26022 
050 
079 
107 
135 

826 

857 

888 

920 

.26951 

.26982 

.27013 

044 

076 

.7277 
.7234 
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.7148 
3.7105 
.7062 
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.6976 
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585 
578 
570 
562 
.96555 
547 
540 
532 
524 

10 

.26163 

.27107 

3.6891 

.96517 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
31 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

11 
i  12 
13 
14 
15 
16 
17 
18 
19 

191 
219 
247 
275 
.26303 
331 
359 
387 
415 

138 
169 
201 
232 
.27263 
294 
326 
357 
388 

.6848 
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3.6680 
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.6596 
.6554 
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509 
502 
494 
486 
.96479 
471 
463 
456 
448 

20 

.26443 

.27419 

3.6470 

.96440 

21 
22 
23 
24 
25 
26 
27 
28 

1  30 

i  ^1 
32 
33 

l34 
35 
36 
37 
38 

i  ^^ 

471 
500 
528 
556 
.26584 
612 
640 
668 
696 

451 
482 
513 
545 
.27576 
607 
638 
670 
701 

.6429 
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.6346 
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3.6264 
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.6181 
.6140 
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433 
425 
417 
410 
.96402 
394 
386 
379 
371 

.26724 

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3.6059 

.96363 

752 
780 
808 
836 

.26864 
892 
920 
948 

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764 

795 

826 

858 

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921 

952 

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.28015 

.6018 
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3.5856 
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355 
347 
340 
332 
.96324 
316 
308 
301 
293 

I  40 

.27004 

.28046 

3.5656 

.96285 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

i  41 
'  42 

i  ^^ 
44 
45 
46 
47 
48 

1  49 
50 

032 
060 
088 
116 
.27144 
172 
200 
228 
256 

077 
109 
140 
172 
.28203 
234 
266 
297 
329 

.5616 
.5576 
.5536 
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3.5457 
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.5339 
.5300 

277 
269 
261 
253 
.96246 
238 
230 
222 
214 

.27284 

.28360 

3.5261 

.96206 

51 
52 
53 
54 
55 
56 
57 
58 
i  59 
60 

312 
340 
368 
396 
.27424 
452 
480 
508 
536 

391 
423 
454 
486 
.28517 
549 
580 
612 
643 

.5222 
.5183 
.5144 
.5105 
3.5067 
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.4989 
.4951 
.4912 

198 
190 
182 
174 
.96166 
158 
150 
142 
134 

.27564 

.28675 

3.4874 

.96126 

N.  Cos. 

N.  Cot. 

N.Tan.|N.  Sin.| 

/ 

12 


75' 


740 


178 

16° 

17° 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

"o" 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

_60_ 

59 

58 
57 

0 

.27564 

.28675 

3.4874 

.96126 

60 

59 
58 
57 

.29237 

.30573 

3.2709 

.95630 

1 

2 
3 

592 
620 
648 

706 
738 
769 

.4836 
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.4760 

118 
110 
102 

1 

2 
3 

265 
293 
321 

605 
637 
669 

.2675 
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.2607 

622 
613 
605 

4 
5 
6 

676 

.27704 
731 

801 

.28832 

864 

.4722 

3.4684 

.4646 

094 

.96086 

078 

56 
55 
54 

4 
5 
6 

348 

.29376 

404 

700 

.30732 

764 

.2573 

3.2539 

.2506 

596 

.95588 

579 

56 
55 
54 

7 

8 

9 

10 

759 

787 
815 

895 
927 
958 

.4608 
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.4533 

070 
062 
054 

53 
52 
51 
50 

49 

48 
47 

7 

8 

9 

10 

432 
460 

487 

796 
828 
860 

.2472 
.2438 
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571 
562 

554 

53 
52 
51 

.27843 

.28990 

3.4495 

.96046 

.29515 

.30891 

3.2371 

.95545 

50 

11 
12 
13 

871 
899 
927 

.29021 
053 
084 

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037 
029 
021 

11 
12 
13 

543 
571 
599 

923 

955 

.30987 

.2338 
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536 
528 
519 

49 

48 
47 

14 
15 
16 

955 

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116 

.29147 

179 

.4346 

3.4308 

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013 
.96005 
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46 
45 
44 

14 
15 
16 

626 

.29654 

682 

.31019 

.31051 

083 

.2238 

3.2205 

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511 

.95502 

493 

46 
45 

44  i 

17 
18 
19 

039 
067 
095 

210 

242 
274 

.4234 
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989 
981 
972 

43 
42 
41 
40 

39 

38 
37 

17 
18 
19 

710 
737 
765 

115 
147 

178 

.2139 
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485 
476 
467 

43 
42 
41 

20 

21 
22 
23 

.28123 

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3.4124 

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20 

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3.2041 

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40 

150 
178 
206 

337 
368 
400 

.4087 
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956 
948 
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21 
22 
23 

821 
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242 
274 
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450 
441 
433 

39 

38 
37 

24 
25 
26 

234 

.28262 

290 

432 

.29463 

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3.3941 

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931 

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36 
35 
34 

24 
25 
26 

904 

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338 

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402 

.1910 

3.1878 
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424 

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407 

36 

35 
34 

27 
28 
29 
30 

318 
346 
374 

526 

558 
590 

.3868 
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907 
898 
890 

33 
32 
31 
30 

27 
28 
29 
30 
31 
32 
33 

.29987 

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043 

.30071 

434 
466 

498 

.1813 
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398 
389 
380 

33 
32 
31 
30 

.28402 

.29621 

3.3759 

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.31530 

3.1716 

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31 
32 
33 

429 

457 
485 

653 
685 
716 

.3723 
.3687 
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874 
865 
857 

29 

28 

27 

098 
126 
154 

562 
594 
626 

.1684 
.1652 
.1620 

363 
354 
345 

29 

28 
27 

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35 
36 

513 

.28541 

569 

748 

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811 

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3.3580 

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849 

.95841 

832 

26 

25 
24 

34 
35 
36 

182 
.30209 

237 

658 
.31690 

722 

.1588 

3.1556 

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337 

.95328 

319 

26 

25 
24 

37 
38 
39 
40 

41 
42 

43 

597 
625 
652 

843 
875 
906 

.3509 

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.3438 

824 
816 
807 

23 
22 
21 
20 

37 
38 
39 

265 
292 
320 

754 
786 
818 

.1492 
.1460 
.1429 

310 
301 
293 

23 

22  1 
21  i 

.28680 
708 
736 
764 

.29938 

3.3402 

.95799 

40 

.30348 

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3.1397 

.95284 

20  ! 
19 
18 
17 

.29970 

.30001 

033 

.3367 
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791 
782 
774 

19 

18 
17 

41 
42 
43 

376 
403 
431 

882 
914 
946 

.1366 
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275 
266 
257 

44 
45 
46 

792 

.28820 

847 

065 

.30097 

128 

.3261 

3.3226 

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766 

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749 

16 
15 
14 

44 
45 
46 

459 

.30486 

514 

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.32010 

042 

.1271 

3.1240 

.1209 

248 

.95240 

231 

16 
15 
14 

47 
48 
49 

875 
903 
931 

160 
192 

224 

.3156 
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.3087 

740 

732 
724 

13 
12 
11 
10 

47 
48 
49 

542 
570 
597 

074 
106 
139 

.1178 
.1146 
.1115 

222 
213 
204 

13 
12 
11 

50 

51 

52 

!  53 

.28959 

.30255 

3.3052 

.95715 

50 

.30625 

.32171 

3.1084 

.95195 

10 

.28987 

.29015 

042 

237 
319 
351 

.3017 
.2983 
.2948 

707 
698 
690 

9 

8 

7 

51 

52 
53 

653 
680 

708 

203 
235 
267 

.1053 
.1022 
.0991 

186 
177 
168 

9 

8 

7 

1  54 
!  55 

:56 

070 

.29098 

126 

382 

.30414 

446 

•  .2914 
3.2879 

.2845 

681 

.95673 

664 

6 

5 
4 

54 
55 
56 

736 

.30763 

791 

299 

.32331 
363 

.0961 

3.0930 

.0899 

159 
.95150 

142 

6 

5 
4 

57 

58 

59 

!  60 

154 

182 
209 

478 
509 
541 

.2811 

.2777 
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656 
647 
639 

3 
2 
1 
0 

57 
58 
59 
60 

819 
846 
874 

396 

428 
460 

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133 
124 
115 

3 
2 

1 

0 

.29237 

.30573 

3.2709 

.95630 

.30902 

.32492 

3.0777 

.95106 

i 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 



N.  Cos. 

N.  Cot.  N.Tan. 

N.  Sin.  ' 

73° 


72' 


lO 


' 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.30902 

.32492 

3.0777 

.95106 

60 

59 

58 
57 

1 

2 
3 

929 

957 

.30985 

524 
556 

588 

.0746 
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097 
088 
079 

4 
5 
6 

.31012 

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068 

621 

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685 

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3.0625 

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070 
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.  052 

56  1 

55 

54 

7 
8 
9 

095 
123 
151 

717 
749 

782 

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043 
033 

024 

53 
52 
51 

10 

.31178 

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3.0475 

.95015 

50 

49 

48 

47  1 

11 
12 
13 

206 
233 
261 

846 
878 
911 

.0445 
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.95006 

.94997 

988 

14 
15 
16 

289 

.31316 

344 

943 
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.33007 

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3.0326 

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979 

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961 

46  i 
45  1 
44  ! 

17 
18 
19 

372 
399 
427 

040 
072 
104 

.0267 
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952 
943 
933 

43  1 
42  1 
41  1 
40 
39  ; 
38  1 
37  , 

20 

.31454 

.33136 

3.0178 

.94924 

21 
22 
23 

482 
510 
537 

169 
201 
233 

.0149 
.0120 
.0090 

915 
906 
897 

24 
25 
26 

565 

.31593 

620 

266 

.33298 

330 

.0061 
3.0032 
3.0003 

888 

.94878 

869 

36  1 
35  ' 
34  j 

27 
28 
29 

648 
675 
703 

363 
395 

427 

2.9974 
.9945 
.9916 

860 
851 
842 

33  ' 
32  1 
31 
30 

30 

.31730 

.33460 

2.9887 

.94832 

31 
32 
33 

758 
786 
813 

492 
524 

557 

.9858 
.9829 
.9800 

823 
814 
805 

29 

28 
27 

34 
35 
36 

841 

.31868 

896 

589 

.33621 

654 

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2.9743 

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795 
.94786 

777 

26 

25  i 
24 

37 
38 
39 

923 

951 

.31979 

686 

718 
751 

.9686 
.9657 
.9629 

768 
758 
749 

23 
22 
21 

40 

.32006 

.33783 

2.9600 

.94740 

20 

19 
18 
17 

41 

42 
43 

034 
061 
089 

816 
848 
881 

.9572 
.9544 
.9515 

730 
721 
712 

44 
45 
46 

116 

.32144 

171 

913 
.33945 
.33978 

.9487 

2.9459 

.9431 

702 

.94693 

684 

16 
15  ; 
14 

47 
48 
49 

199 

227 
254 

.34010 
043 
075 

.9403 
.9375 
.9347 

674 
665 
656 

13! 
12  i 
11  j 
10 

50 

.32282 

.34108 

2.9319 

.94646 

51 
52 
53 

309 
337 
364 

140 
173 
205 

.9291 
.9263 
.9235 

637 
627 
618 

9 

8 

7 

54 
55 
56 

392 
.32419 

447 

238 

.34270 

303 

.9208 

2.9180 

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609 

.94599 

590 

6 

5 
4 

57 
58 
59 

474 
502 
529 

335 
368 
400 

.9125 
.9097 
.9070 

580 
571 
561 

3 
2 
1 
0 

60 

.32557 

.34433 

2.9042 

.94552 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

f    1 

±u 

H\J 

1  ' 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

1 

2 
3 

.32557 

.34433 

2.9042 

.94552 
542 
533 
523 

60 

59 

58 

57 

584 
612 
639 

465 
498 
530 

.9015 
.8987 
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4 
5 
6 

667 
.32694 

722 

563 

.34596 

628 

.8933 

2.8905 

.8878 

514 

.94504 

495 

56 

55 
54 

7 

8 

9 

10 

749 

777 
804 

661 
693 
726 

.8851 
.8824 
.8797 

485 
476 
466 

53 
52 
51 

.32832 

.34758 

2.8770 

.94457 

50 

11 
12 
13 

859 
887 
914 

791 
824 
856 

.8743 
.8716 
.8689 

447 
438 
428 

49 

48 
47 

14 
15 
16 

942 
.32969 
.32997 

889 

.34922 

954 

.8662 

2.8636 

.8609 

418 

.94409 

399 

46 

45 
44 

17 
18 
19 
20 

.33024 
051 
079 

.34987 

.35020 

052 

.8582 
.8556 
.8529 

390 
380 
370 

43 
42 
41 
40 

.33106 

.35085 

2.8502 

.94361 

21 
22 
23 

134 
161 
189 

118 
150 
183 

.8476 
.8449 
.8423 

351 
342 
332 

39 

38 
37 

24 
25 
26 

216 

.33244 

271 

216 

.35248 

281 

.8397 

2.8370 

.8344 

322- 

.94313 

303 

36 
35 
34 

27 
28 
29 
30 

298 
326 

353 

314 
346 
379 

.8318 
.8291 
.8265 

•  293 
284 
274 

33 
32  i 
31  1 
30  1 

.33381 

.35412 

2.8239 

.94264 

i  ^^ 
32 

33 

408 
436 
463 

445 
477 
510 

.8213 
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.8161 

254 
245 
235 

29 

28 
37 

34 
35 
36 

490 

.33518 
545 

543 

.35576 

608 

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2.8109 

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225 

.94215 

206 

26 
25 

24  1 

37 
38 
39 

573 
600 
627 

641 

674 
707 

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196 
186 
176 

23 
22 
21 
20 

40 

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2.7980 

.94167 

41 

42 
43 

682 
710 
737 

772 
805 
838 

.7955 
.7929 
.7903 

157 
147 
137 

19 
18 
17 

44 
45 
46' 

764 

.33792 

819 

871 

.35904 

937 

.7878 

2.7852 

.7827 

127 

.94118 

108 

16 
15 
14 

47 
48 
49 
50 

51 

52 
!  53 

846 
874 
901 

.35969 

.36002 

035 

.7801 
.7776 
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098 

088 
078 

13 
12 
11 

.33929 

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2.7725 

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10 

9 

8 

7 

956 
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.34011 

101 
134 
167 

.7700 
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.7650 

058 
049 
039 

1  54 
55 
56 

038 

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093 

199 

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265 

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2.7600 

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029 
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6 

5 
4 

57 

58 

1  59 

120 
147 
175 

298 
331 
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989 
979 

3 

2  ! 
1  1 
0  1 

60 

.34202 

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2.7475 

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N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

ir 


70° 


180 

20° 

21° 

/ 

N.  Sin. 

N.Tan.N.  Cot. 

N.  Cos. 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

1 

2 
3 

.34202 

.36397 

2.7475 
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.93969 

60 

59 

58 
57 

0 

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2 
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2.6051 
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59 

58 
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229 

2^57 
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430 
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949 
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420 
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487 

348 
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327 

4 
5 
6 

311 

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366 

529 

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595 

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2.7351 

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909 

56 

55 
54 

4 
5 
6 

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520 

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587 

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2.5938 

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316 

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295 

56 

55 
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7 
8 
9 

393 
421 

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628 
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2.7228 

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53 
52 
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50 
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48 
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7 
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027 
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620 
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53 
52 
51 
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10 

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2.5826 
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11 
12 
13 

503 
530 

557 

760 
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11 
12 
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135 
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46 
45 
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14 
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217 

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2.5715 

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46 

45 
44 

17 
18 
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2.6985 

799 
789 
779 

43 
42 
41 
40 

39 

38 

37  j 

17 
18 
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298 
325 

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2.5605 

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169 

159 

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43 
42 
41 
40 

39 

38 
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20 

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.93769 

20 

.36379 

21 
22 
23 

775 
803 
830 

090 
123 

157 

.6961 
.6937 
.6913 

759 
748 
738 

21 
22 
23 

406 
434 
461 

.5583 
.5561 
.5539 

137 
127 
116 

24 
25 
26 

857 

.34884 

912 

190 

.37223 
256 

.6889 

2.6865 

.6841 

728 

.93718 

708 

36  \ 

35 

34 

24 
25 
26 

488 
.36515 

542 

190 
.39223 

257 

.5517 

2.5495 

.5473 

106 

.93095 

084 

36 

35 
34 

27 
28 
29 
30 
31 
32 
33 

939 

966 

.34993 

289 
322 
355 

.6818 
.6794 
.6770 

698 
688 
677 

33 
32 
31  ! 
30 

29 

28 
27 

27 
28 
29 

569 
596 
623 

290 

324 
357 

.5452 
.5430 
.5408 

074 
063 

052 

33 
31 
31 
30 

29 

28 
27 

.35021 

.37388 

2.6746 

.93667 

30 

31 
32 
33 

.36650 

.39391 

2.5386 

.93042 

048 
075 
102 

422 
455 
488 

.6723 
.6699 
.6675 

657 
647 
637 

677 
704 
731 

425 
458 
492 

.5365 
.5343 
.5322 

031 

020 

.93010 

34 
35 
36 

130 

.35157 
184 

521 
.37554 

588 

.6652 

2.6628 

.6605 

626 

.93616 

606 

26  1 
25  • 

24  1 

34 
35 
36 

758 

.36785 

812 

526 

.39559 

593 

.5300 

2.5279 

.5257 

.92999 

.92988 

978 

26 
25 
24 

37 
38 
39 
40 

211 

239 

266 

.35293 

621 
654 

687 

.6581 
.6558 
.6534 

596 

585 
575 

23 
22  1 
21  1 
20 
19  1 
18  1 
17 

37 
38 
39 

839 
867 
894 

626 
660 
694 

.5236 
.5214 
.5193 

967 
956 
945 
.92935 
924 
913 
902 

23 
22 
21 
20 
19 
18 
17 

.37720 

2.6511 

.93565 

555 
544 
534 

40 

41 
42 
43 

.36921 

948 

.36975 

.37002 

.39727 
761 
795 
829 

2.5172 

41 
42 
43 

320 
347 
375 

754 
787 
820 

.6488 
.6464 
.6441 

.5150 
.5129 
.5108 

44 
45 
46 

402 

.35429 

456 

853 

.37887 

920 

.6418 

2.6395 

.6371 

524 

.93514 

503 

16 
15  i 
14 

44 
45 
46 

029 

.37056 
083 

862 

.39896 

930 

.5086 

2.5065 

.5044 

892 

.92881 

870 

16 

15 
14 

47 
48 
49 
50 

51 
52 
53 

484 
511 
538 

953 
.37986 
.38020 

.6348 

.6325 

.6302 

2.6279 

493 

483 
472 

13 
12 

10  j 
9 

8 

7 

47 
48 
49 

110 
137 
164 

963 
.39997 
.40031 

.5023 
.5002 
.4981 

859 
849 

838 

13 
12 
11 
10 

9 

8 

7 

.35565 

.38053 

.93462 
452 
441 
431 

50 

.37191 

.40065 

2.4960 
.4939 
.4918 
.4897 

.92827 

592 
619 
647 

086 
120 
153 

.6256 
.6233 
.6210 

51 

52 
53 

218 
245 
272 

098 
132 
166 

816 
805 
794 

54 
55 
56 

674 

.35701 

728 

186 

.38220 

253 

.6187 

2.6165 

.6142 

420 

.93410 

400 

6 

5 
4 

54 
55 
56 

299 

.37326 
353 

200 

.40234 

267 

.4876 

2.4855 

.4834 

784 

.92773 

762 

6 

5 
4 

57 
58 
59 
60 

755 

782 

810 

.35837 

286 
320 
353 

.6119 

.6096 

.6074 

2.6051 

389 

379 

368 

.93358 

3 
2 
1 
0 

57 
58 
59 
60 

380 

407 

434 

.37461 

301 

335 

369 

.40403 

.4813 
.4792 

.4772 

751 
740 
729 

3 
2 
1 
0 

.38386 

2.4751 

.92718 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

'  1 

• 

N.  Cos. 

N.  Cot.  N.Tan. 

N.  Sin. 

/ 

09° 


68° 


22° 


' 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

60 

59 
58 

57 

0 

.37461 

.40403 

2.4751 
.4730 
.4709 
.4689 

.92718 

1 

2 
3 

488 
515 
542 

436 
470 
504 

707 

697 

.  686 

4 

5 
6 

569 

.37595 

622 

538 

.40572 

606 

.4668 

2.4648 

.4627 

675 

.92664 

653 

56 
55 
54 

7 
8 
9 

649 
676 
703 

640 
674 
707 

.4606 
.4586 
.4566 

642 
631 
620 

53 
52 
51 

10 

.37730 

.40741 

2.4545 

.92609 

50 

49 

48 
47 

11 
12 
13 

757 
784 
811 

775 
809 
843 

.4525 
.4504 
.4484 

598 
587 
576 

14 
15 
16 

838 

.37865 

892 

877 

.40911 

945 

.4464 

2.4443 

.4423 

565 

.92554 

543 

46 

45 
44 

17 
18 
19 

919 
946 
973 

.40979 

.41013 

047 

.4403 
.4383 
.4362 

532 
521 
510 

43 
42 
41 

20 

.37999 

.41081 

2.4342 

.92499 

40 

21 
22 
23 

.38026 
053 
080 

115 
149 
183 

.4322 
.4302 
.4282 

488 
477 
466 

39 
38 
37 

24 
25 
26 

107 

.38134 

161 

217 
.41251 

285 

.4262 

2.4242 
.4222 

455 

.92444 

432 

36 
35 
34 

27 
28 
29 

188 
215 
241 

319 
353 

387 

.4202 
.4182 
.4162 

421 
410 
399 

33 
32 
31 

30 

.38268 

.41421 

2.4142 

.92388 

30 

31 

32 
33 

295 
322 
349 

455 
490 
524 

.4122 
.4102 
.4083 

377 
366 
355 

29 

28 
27 

34 
35 
36 

376 

.38403 

430 

558 

.41592 

626 

.4063 

2.4043 

.4023 

343 

.92332 

321 

26 

25 

24 

37 
38 
39 

456 
483 
510 

660 
694 

728 

.4004 
.3984 
.3964 

310 
299 

287 

23 
22 

21  ! 

40 

.38537 

.41763 

2.3945 

.92276 

20 

41 

42 
43 

564 
591 
617 

797 
831 
865 

.3925 
.3906 
.3886 

265 
254 
243 

19 

18 
17 

44 
45 
46 

644 

.38671 

698 

899 
.41933 
.41968 

.3867 

2.3847 

.3828 

231 

.92220 

209 

16 

15 
14 

47 
48 
49 

725 

752 

778 

.38805 

.42002 
036 
070 

.3808 
.3789 
.3770 

198 
186 
175 

13 
12 
11 
10 

9 

8 

7 

50 

.42105 

2.3750 

.92164 

51 
52 
53 

832 
859 
886 

139 
173 
207 

.3731 
.3712 
.3693 

152 
141 
130 

54 
55 
56 

912 

.38939 

966 

242 

.42276 

310 

.3673 

2.3654 

.3635 

119 

.92107 

096 

6 
5 
4 

57 
58 
59 

.38993 

.39020 

046 

345 
379 
413 

.3616 
.3597 
.3578 

085 

073 

062 

.92050 

3 
2 

-i 

60 

.39073 

.42447 

2.3559 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1    j 

23° 

181 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.39073 

.42447 

2.3559 

.92050 

60 

1 
2 

3 

100 
127 
153 

482 
516 
551 

.3539 
.3520 
.3501 

039 
028 
016 

59 

58 
57 

4 
5 
6 

180 

.39207 

234 

585 

.42619 

654 

.3483 

2.3464 

.3445 

.92005 

.91994 

982 

56 

55 
54 

7 

8 

9 

10 

260 
287 
314 

688 

722 

757 

.42791 

.3426 
.3407 
.3388 

971 
959 
948 

53 
52 
51 
50 

.39341 

2.3369 

.91936 

11 
12 
13 

367 
394 
421 

826 
860 
894 

.3351 
.3332 
.3313 

925 
914 
902 

49 

48 
47 

14 
15 
16 

448 

.39474 

501 

929 
.42963 
.42998 

.3294 

2.3276 

.3257 

891 
.91879 

868 

46 

45 
44 

17 
18 
19 
20 

21 

22 
23 

528 
555 
581 

.43032 
067 
101 

.43136 

.3238 
.3220 
.3201 

856 
845 
833 

43 
42 
41 
40 

39 
38 
37 

.39608 

2.3183 

.91822 

635 
661 

688 

170 
205 
239 

.3164 
.3146 
.3127 

810 
799 

787 

24 
25 
26 

715 

.39741 

768 

274 

.43308 

343 

.3109 

2.3090 

.3072 

775 
.91764 

752 

36 
35 
34 

27 
28 
29 
30 

795 

822 
848 

378 
412 
447 

.3053 
.3035 
.3017 

741 
729 
718 

33 
31 
31 
30 

29 

28 

27 

.39875 

.43481 

2.2998 

.91706 

31 
32 
33 

902 
928 
955 

516 
550 

585 

.2980 
.2962 
.2944 

694 
683 
671 

34 
35 
36 

.39982 

.40008 

035 

620 

.43654 

689 

.2925 

2.2907 

.2889 

660 

.91648 

636 

26 

25 
24 

37 
38 
39 
40 
41 
42 
43 

062 
088 
115 

724 
758 
793 

.2871 
.2853 
.2835 

625 
613 
601 

23 
22 
21 

.40141 

.43828 

2.2817 

.91590 

20 

19 
18 
17 

168 
195 
221 

862 
897 
932 

.2799 
.2781 
.2763 

578 
566 
555 

44 
45 
46 

248 

.40275 

301 

.43966 

.44001 

036 

.2745 

2.2727 

.2709 

543 

.91531 

.  519 

16 
15 
14 

47 
48 
49 

328 
355 
381 

071 
105 
140 

.2691 
.2673 
.2655 

508 
496 

484 

13 
12 
11 

50 

.40408 

.44175 

2.2637 

.91472 

10 

51 

52 
53 

434 
461 

488 

210 
244 
279 

.2620 
.2602 
.2584 

461 
449 
437 

9 

8 

7 

54 
55 
56 

514 

.40541 

567 

314 

.44349 

384 

.2566 

2.2549 

.2531 

425 

.91414 

402 

6 

5 
4 

57 
58 
59 

594 
621 
647 

418 

453 

488 

.44523 

.2513 
.2496 

.2478 

390 
378 
366 

3 
2 
1 
0 

60 

.40674 

2.2460 

.91355 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

182 

24*=* 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

1 

2 
3 

.40674 

.44523 

2.2460 
.2443 
.2425 
.2408 

.91355 
343 
331 
319 

60 

59 

58 

57 

700 

727 
753 

558 
593 
627 

4 
5 
6 

780 

.40806 

833 

662 

.44697 

732 

.2390 

2.2373 

.2355 

307 
.91295 

283 

56 
55 
54 

7 
8 
9 

860 
886 
913 

767 
802 
837 

.2338 
.2320 
.2303 

272 
260 
248 

53 
52 
51 
50 

10 

.40939 

.44872 

2.2286 

.91236 

11 
12 
13 

966 
.40992 
.41019 

907 

942 

.44977 

.2268 
.2251 
.2234 

224 
212 
200 

49 

48 
47 

14 
15 
16 

045 

.41072 

098 

.45012 

.45047 

082 

.2216 

2.2199 

.2182 

188 

.91176 

164 

46 

45 
44 

17 
18 
19 

125 
151 

178 

117 
152 

187 

.2165 
.2148 
.2130 

152 
140 

128 

43 
42 
41 
40 

20 

21 
22 
23 

.41204 

.45222 

2.2113 

.91116 

231 

257 
284 

257 
292 
327 

.2096 
.2079 
.2062 

104 
092 
080 

39 

3S 
37 

24 
25 
26 

310 

.41337 

363 

362 

.45397 

432 

.2045 

2.2028 

.2011 

068 

.91056 

044 

36 
35 
34 

27 
28 
29 
30 
31 
32 
33 

390 
416 
443 

.  467 
502 
538 

.1994 
.1977 
.1960 

032 

020 

.91008 

33 
31 
31 
30 

29 

28 
27 

.41469 

.45573 

2.1943 

.90996 

496 
522 
549 

608 
643 
678 

.1926 
.1909 
.1892 

984 
972 
960 

34 
35 
36 

575 

.41602 

628 

713 

.45748 
784 

.1876 

2.1859 

.1842 

948 

.90936 

924 

26 

25 
24 

37 
38 
39 

655 
681 
707 

819 

854 
889 

.1825 
.1808 
.1792 

911 
899 

887 

23 
22 
21 

40 

41 
42 
43 

.41734 

.45924 

2.1775 

.90875 

20 

19 
18 
17 

760 
787 
813 

960 
.45995 
.46030 

.1758 
.1742 
.1725 

863 
851 
839 

44 
45 
46 

840 

.41866 

892 

065 

.46101 

136 

.1708 

2.1692 

.1675 

826 

.90814 

802 

16 
15 
14 

47 
48 
49 
50 

51 

52 
53 

919 
945 
972 

171 
206 

242 

.1659 
.1642 
.1625 

790 

778 
766 

13 
12 
11 
10 

9 

8 
7 

.41998 

.46277 

2.1609 

.90753 

.42024 
051 
077 

312 
348 
383 

.1592 
.1576 
.1560 

741 
729 
717 

54 
55 
56 

104 

.42130 

156 

418 

.46454 

489 

.1543 

2.1527 
.1510 

704 

.90692 

680 

6 

5 
4 

57 
58 
59 
60 

183 
209 

235 
.42262 

525 
560 
595 

.1494 
.1478 
.1461 

668 
655 
643 

3 
2 
1 
0 

.46631 

2.1445 

.90631 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

25' 


/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.42262 

.46631 
666 
702 
737 

2.1445 

.90631 

60 

59 

58 

57 

1 

2 
3 

288 
315 
341 

.1429 
.1413 
.1396 

618 
606 
594 

4 
5 
6 

367 

.42394 

420 

772 

.46808 

843 

.1380 

2.1364 

.1348 

582 
.90569 

557 

56 

55 
54 

7 
8 
9 

10 
11 
12 
13 

446 
473 
499 

879 
914 
950 

.1332 
.1315 
.1299 

545 
532 
520 

53 
52 
51 
50 
49 
48 
47 

.42525 

.46985 

2.1283 

.90507 

552 
578 
604 

.47021 
056 
092 

.1267 
.1251 
.1235 

495 

483 
470 

14 
15 
16 

631 

.42657 

683 

128 

.47163 

199 

.1219 

2.1203 

.1187 

458 

.90446 

433 

46 
45 
44 

17 
18 
19 
20 

709 
736 
762 

234 

270 

305 

.47341 

.1171 

.1155 

.1139 

2.1123 

421 
408 
396 

43 
42 
41 

.42788 

.90383 

40 

21 
22 
23 

815 
841 
867 

377 
412 

448 

.1107 
.1092 
.1076 

371 
318 
346 

39 
38 
37 

24 
25 
26 

894 

.42920 

946 

483 
.47519 

555 

.1060 

2.1044 

.1028 

334 

.90321 

309 

36 

35 
34 

27 
28 
29 
30 

972 
.42999 
.43025 

590 
626 
662 

.1013 

.0997 

.0981 

2.0965 

296 
284 
271 

33 
31 
31 
30 

29 

28 

27 

.43051 

.47698 

.90259 

31 
32 
33 

077 
104 
130 

733 
769 
805 

.0950 
.0934 
.0918 

246 
233 
221 

34 
35 
36 

156 

.43182 

209 

840 

.47876 

912 

.0903 

2.0887 

.0872 

208 

.90196 

183 

26 

25 
24 

37 
38 
39 
40 

41 
42 
43 

235 
261 

287 

948 
.47984 
.48019 
.48055 

.0856 
.0840 
.0825 

171 

158 
146 

23 
22 
21 
20 

.43313 

2.0809 

.90133 

340 
366 
392 

091 
127 
163 

.0794 
.0778 
.0763 

120 
108 
095 

19 
18 
17 

44 
45 
46 

418 

.43445 

471 

198 

.48234 
270 

.0748 

2.0732 

.0717 

082 

.90070 

057 

16 
15 
14 

47 
48 
49 

497 

523 
549 

306 

342 
378 

.0701 
.0686 
.0671 

045 
032 
019 
.90007 
.89994 
981 
968 

13 
12 
11 
10 

9 

8 

7 

50 

.43575 

.48414 

2.0655 

51 

52 
53 

602 
628 
654 

450 
486 
521 

.0640 
.0625 
.0609 

54 
55 
56 

680 

.43706 

733 

557 

.48593 

629 

.0594 

2.0579 

.0564 

956 

.89943 

930 

6 

5 
4 

57 
58 
59 
60 

759 

785 
811 

665 
701 

737 

.0549 
.0533 
.0518 

918 
905 
892 

3 
2 
1 
0 

.43837 

.48773 

2.0503 

.89879 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

'. 

26° 

27° 

183 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

~o" 

1 

2 
3 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.43837 

.48773 

2.0503 

.89879 

60 

.45399 

.50953 

1.9626 

.89101 

60 

1 

2 
3 

863 
889 
916 

809 
845 
881 

.0488 
.0473 
.0458 

867 
854 
841 

59 

58 
57 

425 
451 
477 

.50989 

.51026 

063 

.9612 
.9598 
.9584 

087 
074 
061 

59 

58 
57 

4 
5 
6 

942 
.43968 
.43994 

917 

.48953 
.48989 

.0443 

2.0428 

.0413 

828 

.89816 

803 

56 

55 
54 

4 
5 
6 

503 
.45529 

554 

099 

.51136 

173 

.9570 

1.9556 

.9542 

048 

.89035 

021 

56 

55 
54 

7 

8 

9 

10 

.44020 
046 
072 

.49026 
062 
098 

.0398 
.0383 
.0368 

790 

777 
764 

53 
52 
51 

7 
8 
9 

580 
606 
632 

209 
246 
283 

.9528 
.9514 
.9500 

.89008 

.88995 

981 

53 
52 
51 

.44098 

.49134 

2.0353 

.89752 

50 

10 

11 
12 
13 

.45658 

.51319 

1.9486 

.88968 

50 

11 
12 
13 

124 
151 
177 

170 
206 
242 

.0338 
.0323 
.0308 

739 
726 
713 

49 

48 
47 

684 
710 
736 

356 
393 
430 

.9472 
.9458 
.9444 

955 
942 
928 

49 

48 
47 

14 
15 
16 

203 
.44229 

255 

278 

.49315 

351 

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2.0278 

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700 

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674 

46 

45 
44 

14 
15 
16 

762 

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813 

467 

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540 

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1.9416 

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915 

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46 
45 
44 

17 
18 
19 

281 
307 
333 

387 
423 
459 

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662 
649 
636 

43 
42 
41 

17 
18 
19 

839 
865 
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577 
614 
651 

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875 
862 
848 

43 
42 
41 
40 

20 

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2.0204 

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40 

1  20 

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1.9347 

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21 
22 
23 

385 
411 
437 

532 
568 
604 

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610 
597 

584 

39 
38 
37 

21 

22 
23 

942 

968 

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724 
761 
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808 
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39 

38 
37 

24 
25 
26 

464 

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516 

640 

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713 

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2.0130 

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36 

35 
34 

24 
25 
26 

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835 

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909 

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1.9278 

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782 

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36 

35 
34 

27 
28 
29 
30 
31 
32 
33 

542 
568 
594 

749 

786 
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532 
519 
506 

33 
32 
31 
30 

27 

1  28 

29 

30 

097 
123 
149 

946 

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741 

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715 

33 
32 
31 

.44620 

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2.0057 

.89493 

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1.9210 

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30 

646 
672 
698 

894 

931 

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2.0013 

480 
467 
454 

29 

28 
27 

31 
32 
33 

201 
226 
252 

094 
131 
168 

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674 
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29 
28 
27 

34 
35 
36 

724 

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1.9999 

1.9984 

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441 

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415 

26 
25 
24 

34 
35 
36 

278 

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330 

205 

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279 

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1.9142 

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647 

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620 

26 
25 
24 

37 
38 
39 

802 
828 
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113 
149 

185 

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402 
389 
376 

23 
22 
21 

37 
38 
39 
40 

355 

381 

407 

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316 
353 
390 

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607 
593 

580 

23 
22 
21 

40 

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1.9912 

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20 

19 

18 
17 

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1.9074 

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20 

41 
42 
43 

906 
932 
958 

258 
295 
331 

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350 
337 
324 

41 
42 
43 

458 
484 
510 

464 
501 
538 

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553 
539 
526 

19 
18 
17 

44 
45 
46 

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036 

368 

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441 

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1.9840 

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311 
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285 

16 
15 
14 

44 
45 
46 

536 
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587 

575 

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1.9007 

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512 

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485 

16 
15 
14 

47 
48 
49 

062 
088 
114 

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514 
550 

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272 
259 
245 

13 
12 
11 

47 
48 
49 

613 
639 
664 

687 
724 
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13 
12 
11 
10 

50 

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1.9768 

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10 

50 

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1.8940 

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51 
52 
53 

166 
192 
218 

623 
660 
696 

.9754 
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219 
206 
193 

9 

8 

7 

51 
52 
53 

716 

742 
767 

836 
873 
910 

.8927 
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417 
404 
390 

9 

8 
7 

54 
55 
56 

243 

.45269 

295 

733 

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806 

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1.9697 

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180 

.89167 

153 

6 

5 
4 

54 
55 
56 

793 
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844 

947 

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1.8873 

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377 

.88363 

349 

6 

5 
4 

57 
58 
59 

321 

347 
373 

843 
879 
916 

.9669 
.9654 
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140 
127 
114 

3 
2 
1 

57 
58 
59 
60 

870 
896 
921 

059 
096 

134 

.8847 
.8834 
.8820 

336 

322 
308 

3 
2 

1 

60 

.45399 

.50953 

1.9626 

.89101 

0 

.46947 

.53171 

1.8807 

.88295 

0 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

N.  Cos. 

N.  Cot. 

N.  Tan. 

N.  Sin. 

/ 

184 

28° 

29° 

1 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.46947 

.53171 

1.8807 
.8794 
.8781 
.8768 

.88295 

60 

59 

58 
57 

0 

.48481 

.55431 

1.8040 

.87462 

60 

1 

2 
3 

973 
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208 
246 
283 

281 
267 
254 

1 

2 
3 

506 
532 
557 

469 

507 
545 

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448 
434 
420 

59  ! 
58 

57 

4 
5 
6 

050 

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101 

320 

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395 

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1.8741 

.8728 

240 

.88226 

213 

56 

55 
54 

4 
5 
6 

583 

.48608 

634 

583 

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659 

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1.7979 

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406 

.87391 

377 

56 

55 
54 

7 
8 
9 

127 
153 
178 

432 
470 
507 

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199 
185 
172 

53 
52 
51 
50 
49 
48 
47 

7 

8 

9 

10 

659 
684 
710 

697 
736 

774 

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363 
349 
335 

53 
52 
51 

10 

.47204 

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1.8676 

.88158 

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1.7917 

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50 

49 

48 
47 

11 
12 
13 

229 
255 
281 

582 
620 
657 

.8663 
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144 
130 
117 

11 
12 
13 

761 
786 
811 

850 
888 
926 

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306 
292 
278 

14 
15 
16 

306 

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358 

694 

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769 

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1.8611 

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103 

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075 

46 

45 
44 

14 
15 
16 

837 

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888 

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041 

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1.7856 

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264 

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235 

46 

45 
44 

17 
18 
19 

383 
409 
434 

807 

844 
882 

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062 
048 
034 

43 
42 
41 

17 
18 
19 

913 
938 
964 

079 
117 
156 

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221 
207 
193 

43 
42 
41 

20 

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1.8546 

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40 

39 
38 
37 

20 

21 

22 
23 

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1.7796 

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40 

39 

38 
37 

21 
22 
23 

486 
511 

537 

957 
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979 

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040 
065 

232 
270 
309 

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164 
150 
136 

24 
25 
26 

562 

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614 

070 

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145 

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1.8482 

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965 

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937 

36 
35 
34 

24 
25 
26 

090 

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141 

347 

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424 

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1.7735 

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121 

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093 

36 

35 
34 

27 
28 
29 

639 
665 
690 

183 
220 

258 

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909 
896 

33 
31 
31 

• 

27 
28 
29 
30 

166 
192 
217 

462 
501 
539 

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079 
064 
050 

33 
31 
31 
30 

29 

28 
27 

30 

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1.8418 

8.7882 

30 

29 

28 

27 

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1.7675 

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31 
32 
2,3 

741 
767 
793 

333 
371 
409 

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868 
854 
840 

31 
32 
33 

268 
293 
318 

616 
654 
693 

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021 
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34 
35 
36 

818 

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869 

446 

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522 

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1.8354 

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826 

8.7812 

798 

26 

25 
24 

34 
35 
36 

344 

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394 

731 

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1.7615 

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978 

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949 

26 

25 
24 

37 
38 
39 
40 

41 
42 

43 

895 
920 
946 

560 
597 
635 

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784 
770 
756 

23 
22 
21 

37 
38 
39 

419 

445 
470 

846 
885 
923 

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935 
921 
906 

23 
22 
21 

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1.8291 

8.7743 

20 

40 

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1.7556 

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20 

.47997 

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048 

711 

748 
786 

.8278 
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729 
715 
701 

19 

18 
17 

41 
42 
43 

521 
546 
571 

.57000 
039 
078 

.7544 
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878 
863 
849 

19 

18 
17 

44 
45 
46 

073 

.48099 

124 

824 

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900 

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1.8228 

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687 

8.7673 

659 

16 
15 
14 

44 
45 
46 

596 

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647 

116 

.57155 
193 

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1.7496 

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834 

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805 

16 
15 
14 

47 
48 
49 
50 

150 

175 

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938 
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645 
631 
617 

13 
12 
11 
10 

9 

8 

7 

47 
48 
49 

672 
697 

723 

232 

271 

309 

.57348 

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.7449 

791 

777 
762 

13 
12 
11 

.48226 

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1.8165 

.87603 

50 

.49748 

1.7437 

.86748 

10 

51 

52 
53 

252 
277 
303 

089 
127 
165 

.8152 
.8140 
.8127 

589 
575 
561 

51 
52 
53 

773 
798 
824 

386 

425 
464 

.7426 
.7414 
.7402 

733 
719 
704 

9 

8 

7 

54 
55 
56 

328 

.48354 

379 

203 

.55241 

279 

.8115 

1.8103 

.8090 

546 

.87532 
518 

6 

5 
4 

54 
55 
56 

849 

.49874 

899 

503 

.57541 

580 

.7391 

1.7379 

.7367 

690 

.86675 

661 

6 

5 
4 

57 
58 
59 
60 

405 
430 
456 

317 
355 
393 

.8078 
.8065 
.8053 

504 
490 
476 

3 
2 
1 
0 

57 
58 
59 

924 

950 

.49975 

619 
657 
696 

.7355 
.7344 
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646 
632 
617 

3 
2 
1 
0 

.48481 

.55431 

1.8040 

.87462 

60 

.50000 

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1.7321 

.86603 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

t 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

30° 

/ 

N.  Sin. 

N.Tan.'N.  Cot. 

N.  Cos. 

1 

0 

1 

.50000 

.57735  1  1.7321 

.86603 

60 

025 

774 

.7309 

588 

59 

2 

050 

813 

.7297 

573 

58 

3 

076 

851 

.7286 

559 

57 

4 

101 

890 

.7274 

544 

56  j 

5 

.50126 

.57929 

1.7262 

.86530 

55  1 

6 

151 

.57968 

.7251 

515 

54 

7 

176 

.58007 

.7239 

501 

53 

8 

201 

046 

.7228 

486 

52 

9 

227 

085 

.7216 

471 

51 

10 

.50252 

.58124 

1.7205 

.86457 

50 

11 

277 

162 

.7193 

442 

49 

12 

302 

201 

.7182 

427 

48 

13 

327 

240 

.7170 

413 

47 

14 

352 

279 

.7159 

398 

46 

15 

.50377 

.58318 

1.7147 

.86384 

45 

16 

403 

357 

.7136 

369 

44 

17 

428 

396 

.7124 

354 

43 

18 

453 

435 

.7113 

340 

42 

19 
20 

21 

478 

474 

.7102 

325 

41 
40 

.50503 

.58513 

1.7090 

.86310 

528 

552 

.7079 

295 

39 

22 

553 

591 

.7067 

281 

38 

23 

578 

631 

.7056 

266 

37 

24 

603 

670 

.7045 

251 

36 

25 

.50628 

.58709 

1.7033 

.86237 

35 

26 

654 

748 

.7022 

222 

34 

27 

679 

787 

.7011 

207 

33 

28 

704 

826 

.6999 

192 

32 

29 

729 

865 

.6988 

178 

31 

30 

31 

.50754 

.58905  1  1.6977 

.86163 

30 

779 

944 

.6965 

148 

29 

32 

804 

.58983 

.6954 

133 

28 

33 

829 

.59022 

.6943 

119 

27 

34 

854 

061 

.6932 

104 

26 

35 

.50879 

.59101 

1.6920 

.86089 

25 

36 

904 

140 

.6909 

074 

24 

37 

929 

179 

.6898 

059 

23 

38 

954 

218 

.6887 

045 

22 

39 
40 

.50979 

258 

.6875 

030 

21 
20 

19 

.51004 

.59297 

1.6864 

.86015 

41 

029 

336 

.6853 

.86000 

42 

054 

376 

.6842 

.85985 

18 

43 

079 

415 

.6831 

970 

17 

44 

104 

454 

.6820 

.  956 

16 

45 

.51129 

.59494 

1.6808 

.85941 

15 

46 

154 

533 

.6797 

926 

14 

47 

179 

573 

.6786 

911 

13 

48 

204 

612 

.6775 

896 

12 

49 
50 

229 

651 

.6764 

881 

11 

.51254 

.59691 

1.6753 

.85866 

10 

51 

279 

730 

.6742 

851 

9 

52 

304 

770 

.6731 

836 

8 

53 

329 

809 

.6720 

821 

7 

54 

354 

849 

.6709 

806 

6 

55 

.51379 

.59888 

1.6698 

.85792 

5 

56 

404 

928 

.6687 

777 

4 

57 

429 

.59967 

.6676 

762 

3 

58 

454 

.60007 

.6665 

747 

2 

59 

479 

046 

.6654 

732 

1 

60 

.51504 

.60086 

1.6643 

.85717 

0 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

/ 

31° 

185 

f 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.51504 

.60086 

1.6643 

.85717 

60 

1 

2 
3 

529 
554 
579 

126 
165 
205 

.6632 
.6621 
.6610 

702 
687 
672 

59 

58 
57 

4 
5 
6 

604 

.51628 

653 

245 

.60284 

324 

.6599 

1.6588 

.6577 

657 

.85642 

627 

56 
55 
54 

7 

8 

9 

10 

678 
703 

728 

364 
403 

443 

.6566 
.6555 
.6545 

612 
597 

582 

53 
52 
51 

.51753 

.60483 

1.6534 

.85567 

50 

i  11 

i  12 

13 

778 
803 
828 

522 
562 
602 

.6523 
.6512 
.6501 

551 
536 
521 

49 

48 
47 

14 
15 
16 

852 

.51877 

902 

642 

.60681 

721 

.6490 

1.6479 

.6469 

506 

.85491 

476 

46 
45 
44 

17 
18 
19 

927 

952 

.51977 

761 
801 
841 

.6458 
.6447 
.6436 

461 
446 
431 

43 
42 
41 

20 

.52002 

.60881 

1.6426 

.85416 

40 

21 

22 
23 

026 
051 
076 

921 
.60960 
.61000 

.6415 
.6404 
.6393 

401 
385 
370 

39 

38 
37 

24 
25 
26 

101 

.52126 

151 

040 

.61080 

120 

.6383 

1.6372 

.6361 

355 

.85340 

325 

36 
35 
34 

27 
28 
29 

175 
200 

225 

160 
200 
240 

.6351 
.6340 
.6329 

310 
294 
279 

33 
32 
31 

30 

.52250 

.61280 

1.6319 

.85264 

30 

31 
32 
33 

275 
299 
324 

320 
360 
400 

.6308 
.6297 
.6287 

249 
234 
218 

29 

28 
27 

34 
35 
36 

349 

.52374 
399 

440 

.61480 

520 

.6276 

1.6265 

.6255 

203 

.85188 

173 

26 
25 
24 

37 
3S 
39 
40 

423 
448 
473 

561 
601 
641 

.6244 
.6234 
.6223 

157 
142 
127 

23 
22 
21 

.52498 

.61681 

1.6212 

.85112 

20 

41 

42 
43 

522 
547 
572 

721 
761 
801 

.6202 
.6191 
.6181 

096 
081 
066 

19 
18 
17 

44 
45 
46 

597 

.52621 

646 

842 

.61882 

922 

.6170 

1.6160 

.6149 

051 

.85035 

020 

16 
15 
14 

47 
48 
49 

671 
696 

720 

.61962 

.62003 

043 

.6139 
.6128 
.6118 

.85005 

.84989 

974 

13 
12 
11 

50 

.52745 

.62083 

1.6107 

.84959 

10 

51 

52 
53 

770 
794 
819 

124 
164 
204 

.6097 
.6087 
.6076 

943 
928 
913 

9 

8 

7 

54 
55 
56 

844 

.52869 

893 

245 

.62285 

325 

.6066 

1.6055 

.6045 

897 

.84882 

866 

6 

5 
4 

57 
58 
59 

918 
943 
967 

366 
406 
446 

.6034 
.6024 
.6014 

851 
836 
820 

3 
2 

1 

60 

.52992 

.62487 

1.6003 

.84805 

0 

1 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

186 

32° 

33° 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

60 

59 

58 
57 

/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.52992 

.62487 

1.6003 

.84805 

0 

.54464 

.64941 

1.5399 

.83867  60 

1 

2 
3 

.53017 
041 
066 

527 
568 
608 

.5993 
.5983 
.5972 

789 
774 
759 

1 

2 
3 

488 
513 

537 

.64982 

.65024 

065 

.5389 
.5379 
.5369 

851 
835 
819 

59 

58 
57 

4 
5 
6 

091 

.53115 

140 

649 

.62689 

730 

.5962 

1.5952 

.5941 

743 

.84728 

712 

56 

55 
54 

4 
5 
6 

561 

.54586 

610 

106 

.65148 

189 

.5359 

1.5350 

.5340 

804 
.83788 

772 

56 

55 

54 

7 
8 
9 

164 
189 
214 

770 
811 

852 

.5931 
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697 
681 
666 

53 
52 
51 

7 
8 
9 

635 
659 
683 

231 

272 
314 

.5330 
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756 
740 
724 

53 
52 
51 
50 

10 

.53238 

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1.5900 

.84650 

50 

10 

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1.5301 

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11 
12 
13 

263 
288 
312 

933 
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635 
619 
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49 

48 
47 

11 
12 
13 

732 
756 
781 

397 

438 
480 

.5291 

.5282 
.5272 

692 
676 
660 

49 

48 
47 

14 
15 
16 

337 

.53361 

386 

055 

.63095 

136 

.5859 

1.5849 

.5839 

588 
.84573 

557 

46 

45 
44 

14 
15 
16 

805 

.54829 

854 

521 

.65563 

604 

.5262 

1.5253 

.5243 

645 

.83629 

613 

46 
45 
44 

17 
18 
19 

411 
435 
460 

177 

217 
258 

.5829 
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542 
526 
511 

43 
42 
41 

17 
18 
19 

878 
902 
927 

646 
688 
729 

.5233 
.5224 
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597 

581 

565 

.83549 

43 
42 
41 
40 
39 
38 
37 

20 

.53484 

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1.5798 

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40 

20 

.54951 

.65771 

1.5204 

21 
22 
23 

509 
534 
558 

340 
380 
421 

.5788 
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480 
464 
448 

39 

38  j 
37 

21 
22 
23 

975 
.54999 
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813 
854 
896 

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533 
517 
501 

24 

25 
26 

583 

.53607 

632 

462 

.63503 

544 

.5757 

1.5747 

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433 

.84417 

402 

36  1 

35 

34 

24 
25 
26 

048 

.55072 

097 

938 
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.5166 

1.5156 

.5147 

485 

.83469 

453 

36 

35 
34 

27 
28 
29 
30 

656 
681 
705 

584 
625 
666 

.5727 
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386 
370 

355 

33 
32 
31 

27 
28 
29 

121 
145 
169 

063 
105 
147 

.5137 
.5127 
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437 
421 
405 

33 
31 
31 
30 

.53730 

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1.5697 

.84339 

30 

30 

.55194 

.66189 

1.5108 

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31 
32 
33 

754 
779 
804 

748 
789 
830 

.5687 
.5677 
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324 
308 
292 

29 

28 

27 

31 
32 
33 

218 
242 
266 

230 
272 
314 

.5099 
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373 
356 
340 

29 

28 
27 

34 
35 
36 

828 

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877 

871 

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953 

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1.5647 

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277 

.84261 

245 

26 

25 
24 

34 
35 
36 

291 

.55315 

339 

356 

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440 

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1.5061 

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324 

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292 

26 

25 
24 

37 
38 
39 

902 
926 
951 

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076 

.5627 
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230 
214 
198 

23 
22 
21 
20 
19 
18 
17 

37 
38 
39 

363 

388 
412 

482 
524 
566 

.5042 
.5032 
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276 
260 
244 

23 
22 
21 

40 

41 
42 
43 

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.64117 

1.5597 

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40 

.55436 

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1.5013 

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20 

19 
18 
17 

.54000 
024 
049 

158 
199 
240 

.5587 
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167 
151 
135 

41 

42 
43 

460 
484 
509 

650 
692 

734 

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212 
195 
179 

44 
45 
46 

073 

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122 

281 

.64322 

363 

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1.5547 

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120 

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088 

16 
15 
14 

44 
45 
46 

533 

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581 

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860 

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1.4966 

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163 

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131 

16 
15 
14 

47 
48 
49 
50 

51 

52 
53 

146 
171 
195 

404 
446 
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057 
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13 
12 
11 
10 

47 
48 
49 

605 
630 

654 

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944 

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1.4919 

115 
098 
082 

13 
12 
11 
10 

9 

8 

7 

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.64528 

1.5497 

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50 

.55678 

.67028 

.83066 

244 
269 
293 

569 
610 
652 

.5487 
.5477 
.5468 

.84009 

.83994 

978 

9 

8 

7 

51 

52 
53 

702 
726 
750 

071 
113 
155 

.4910 
.4900 
.4891 

050 
034 
017 

54 
55 
56 

317 

.54342 

366 

693 

.64734 

775 

.5458 

1.5448 

.5438 

962 

.83946 

930 

6 

5 
4 

54 
55 
56 

775 

.55799 

823 

197 
.67239 

282 

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1.4872 

.4863 

.83001 

.82985 

969 

6 

5 
4 

57 
58 
59 

391 

415 
440 

817 
858 
899 

.5428 
.5418 
.5408 

915 
899 

883 

3 
2 
1 

57 
58 
59 

847 
871 
895 

324 
366 
409 

.4854 
.4844 
.4835 

953 

936 

920 

.82904 

3 
2 
1 
0 

60 

.54464 

.64941 

1.5399 

.83867 

0 

60 

.55919 

.67451 

1.4826 

(n.  Cos. 

N.  Cot.  N.Tan. 

N.  Sin. 

r 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

' 

34° 

35° 

187 

/ 
0 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

i 

' 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

"eo" 

59 

58 
57 

.55919 

.67451 

1.4826 

.82904 

60  j 

59 

58 

57  1 

0 

1 

2 
3 

.57358 

.70021 

1.4281 

.81915 

1 

2 
3 

943 

968 

.55992 

493 
536 

578 

.4816 
.4807 
.4798 

887 
871 
855 

381 
405 
429 

064 
107 
151 

.4273 
.4264 
.4255 

899 
882 
865 

4 
5 
6 

.56016 

.56040 

064 

620 

.67663 

705 

.4788 

1.4779 

.4770 

839 

.82822 

806 

56  : 

55  1 
54  i 

4 
5 
6 

453 

.57477 
501 

194 

.70238 
281 

.4246 

1.4237 

.4229 

848 

.81832 

815 

56 

55 
54 

7 
8 
9 

088 
112 
136 

748 
790 
832 

.4761 
.4751 

.4742 

790 

773 

757 

53 
52 
51 

7 

8 

9 

10 

524 
548 
572 

325 
368 
412 

.4220 
.4211 
.4202 

798 
782 
765 

53 
52 
51 
50 
49 
48 
47 

10 

11 
12 
13 

.56160 

.67875 

1.4733 

.82741 

50 

49 

48 
47 

.57596 

.70455 

1.4193 

.81748 

184 
208 
232 

917 
.67960 
.68002 

.4724 
.4715 
.4705 

724 
708 
692 

11 
12 
13 

619 
643 
667 

499 

542 
586 

.4185 
.4176 
.4167 

731 
714 
698 

14 
15 
16 

256 

.56280 

305 

045 

.68088 

130 

.4696 

1.4687 

.4678 

675 

.82659 

643 

46 
45 
44 

14 
15 
16 

691 

.57715 
738 

629 

.70673 

717 

.4158 

1.4150 

.4141 

681 

.81664 

647 

46 
45 
44 

17 
18 
19 

329 
353 

377 

173 
215 

258 

.4669 
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626 
610 
593 

.82577 

43 
42 
41 
40 

17 
18 
19 
20 

762 
786 
810 

760 

804 

848 

.70891 

.4132 
.4124 
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631 
614 
597 

43 
42 
41 
40 
39 
38 
37 

20 

.56401 

.68301 

1.4641 

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1.4106 

.81580 

21 
22 
23 

425 

'  449 

473 

343 
386 
429 

.4632 
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561 

544 
528 

39 

38 
37 

21 
22 
23 

857 
881 
904 

935 
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563 
546 
530 

24 
25 
26 

497 

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545 

471 
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557 

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1.4596 

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511 
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478 

36 
35 
34 

24 
25 
26 

928 

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976 

066 
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154 

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1.4063 

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513 

.81496 

479 

36 

35 
34 

27 
28 
29 

569 
593 
617 

600 
642 
685 

.4577 

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1.4550 

462 
446 
429 

33 

32 
31 

27 
28 
29 

.57999 

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047 

198 
242 

285 

.4045 
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462 

445 
428 

33 
31 
31 
30 

30 

31 
32 
33 

.56641 

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.82413 

30 

30 

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1.4019 

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665 
689 
713 

771 
814 

857 

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396 
380 
363 

29 

28 

27 

31 

32 
33 

094 
118 
141 

373 
417 
461 

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395 
378 
361 

29 

28 
27 

34 
35 
36 

736 

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784 

900 
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1.4505 

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347 

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314 

26 

25 
24 

34 
35 
36 

165 

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212 

505 

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593 

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1.3976 

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344 

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310 

26 

25 
24 

37 
38 
39 

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832 
856 

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071 
114 

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297 
281 
264 

23 
22 
21 

37 
38 
39 

236 
260 
283 

637 
681 

725 

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1.3934 

293 
276 
259 

23 
22 
21 
20 
19 
18 
17 

40 

41 

42 
43 

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1.4460 

.82248 

20 

40 

.58307 

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904 
928 
952 

200 
243 
286 

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231 
214 
198 

19 

18 
17 

41 
42 
43 

330 
354 
378 

813 
857 
901 

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225 
208 
191 

44 
45 
46 

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024 

329 

.69372 

416 

.4424 

1.4415 

.4406 

181 

.82165 

148 

16 
15 
14 

44 
45 
46 

401 

.58425 
449 

946 
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.72034 

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1.3891 

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174 

.81157 

140 

16 
15 
14 

47 
48 
49 
50 

047 
071 
095 

459 
502 
545 

.4397 
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.4379 

132 
115 
098 

13 
12 
11 
10 

47 
48 
49 

472 
496 
519 

078 
122 
167 

.3874 
.3865 
.3857 

123 
106 
089 

13 
12 
11 
10 

9 

8 

7 

.57119 

.69588 

1.4370 

.82082 

50 

51 

52 
53 

.58543 

.72211 

1.3848 

.81072 

51 

52 
53 

143 
167 
191 

631 
675 

718 

.4361 

.4352 
.4344 

065 
048 
032 

9 

8 

7 

567 
590 
614 

255 
299 
344 

.3840 
.3831 
.3823 

.  055 
038 
021 

54 
55 
56 

215 

.57238 

262 

761 

.69804 

847 

.4335 

1.4326 

.4317 

.82015 

.81999 

982 

6 

5 
4 

54 
55 
56 

637 

.58661 

684 

388 
.72432 

477 

.3814 

1.3806 

.3798 

.81004 

.80987 

970 

6 

5 
4 

57 
58 
59 
60 

286 
310 

334 

891 

934 

.69977 

.4308 

.4299 

.4290 

1.4281 

965 
949 
932 

3 
2 
1 

57 
58 
59 

708 
731 
755 

521 
565 
610 

.3789 
.3781 
.3772 

953 
936 
919 

3 
2 
1 
0  1 

.57358 

.70021 

.81915 

0 

60 

.58779 

.72654 

1.3764 

.80902 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

' 

1 
1 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

t 

188 

36° 

1 

N.  Sin.  N.  Tan. 

N.  Cot. 

N.  Cos. 

0 

.58779  1.72654 

1.3764 

.80902 

60 

2 
3 

802 
826 
849 

699 

743 
788 

.3755 
.3747 
.3739 

885 
867 
850 

59 

58 

57 

4 

5 
6 

873 

.58896 

920 

832 

.72877 

921 

.3730 

1.3722 

.3713 

833 

.80816 

799 

56 

55 
54 

7 
8 
9 
10 
11 
12 
13 

943 

967 

.58990 

.72966 

.73010 

055 

.3705 
.3697 
.3688 

782 
765 
748 

53 

52 
51 

.59014 

.73100 

1.3680 

.80730 
713 
696 
679 

50 

49 

48 
47 

037 
061 
084 

144 
189 
234 

.3672 
.3663 
.3655 

14 
15 
16 

108 

.59131 

154 

278 

.73323 

368 

.3647 

1.3638 

.3630 

662 

.80644 

627 

46 
45 
44 

17 
18 
19 
20 

1  21 
22 
23 

178 
201 
225 

413 

457 
502 

.3622 
.3613 
.3605 

610 

593 

576 

.80558 

43 
42 
41 
40 

.59248 

.73547 

1.3597 

272 
295 
318 

592 
637 
681 

.3588 
.3580 
.3572 

541 
524 
507 

39 

37 

24 
25 
26 

342 

.59365 

389 

726 

.73771 

816 

.3564 

1.3555 

.3547 

489 

.80472 

455 

36 

35 
34 

27 
28 
29 
30 
31 
32 

412 
436 
459 

861 
906 
951 

.3539 
.3531 
.3522 

438 
420 
403 

32 
31 

.59482 

.73996 

1.3514 

.80386 

30 

506 
529 

552 

.74041 
086 
131 

.3506 
.3498 
.3490 

368 
351 
334 

29 

28 

27 

34 
35 
36 

576 

.59599 

622 

176 

.74221 

267 

.3481 

1.3473 

.3465 

316 
.80299 

282 

26 

25 
24 

37 
38 
39 
40 

646 
669 
693 

312 
357 
402 

.3457 
.3449 
.3440 

264 
247 
230 

23 
22 
21 

.59716 

.74447 

1.3432 

.80212 

20 

41 
42 
43 

739 

763 
786 

492 

538 
583 

.3424 
.3416 
.3408 

195 
178 
160 

19 

18 
17 

44 
45 
46 

809 

.59832 

856 

628 

.74674 

719 

.3400 

1.3392 

.3384 

143 

.80125 

108 

16 
15 
14 

47 
48 
49 
50 

51 

52 
53 

879 
902 
926 

764 
810 

855 

.3375 
.3367 
.3359 

091 
073 
056 

13 
12 
11 

.59949 

.74900 

1.3351 

.80038 

10 

972 
.59995 
.60019 

946 
.74991 
.75037 

.3343 
.3335 
.3327 

021 
.80003 
.79986 

9 

8 

7 

54 
55 
56 

042 

.60065 

089 

082 

.75128 

173 

.3319 

1.3311 

.3303 

968 

.79951 

934 

6 

5 
4 

57 
58 
59 

112 
135 
158 

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264 
310 

.3295 

.3287 
.3278 

916 
899 

881 

3 
2 
1 

60 

.60182 

.75355 

1.3270 

.79864 

0 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin.  '     1 

37 


/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

60 

59" 

0 

.60182 

.75355 

1.3270 

.79864 

1 

205 

401 

.3262 

846 

2 

228 

447 

.3254 

829 

58 

3 

251 

492 

.3246 

811 

57 

4 

274 

538 

.3238 

793 

56 

5 

.60298 

.75584 

1.3230 

.79776 

55 

6 

321 

629 

.3222 

758 

54 

7 

344 

675 

.3214 

741 

53 

8 

367 

721 

.3206 

723 

52 

9 

390 

767 

.3198 

706 

51 

10 

.60414 

.75812 

1.3190 

.79688 

50 

11 

437 

858 

.3182 

671 

49 

12 

460 

904 

.3175 

653 

48 

13 

483 

950 

.3167 

635 

47 

14 

506 

.75996 

.3159 

618 

46 

15 

.60529 

.76042 

1.3151 

.79600 

45 

16 

553 

088 

.3143 

583 

44 

17 

576 

134 

.3135 

565 

43 

18 

599 

180 

.3127 

547 

42 

19 
20 

21" 

622 

226 

.3119 

530 
.79512 

41 
40 

39 

.60645 

.76272 

1.3111 

668 

318 

.3103 

494 

22 

691 

364 

.3095 

477 

ZZ 

23 

714 

410 

.3087 

459 

37 

24 

738 

456 

.3079 

441 

36 

25 

.60761 

.76502 

1.3072 

.79424 

35 

26 

784 

548 

.3064 

406 

34 

27 

807 

594 

.3056 

388 

ZZ 

28 

830 

640 

.3048 

371 

32 

29 

853 

686 

.3040 

353 

.79335 

318 

31 
30 

29 

30 

31 

.60876 

.76733 

1.3032 

899 

779 

.3024 

32 

922 

825 

.3017 

300 

28 

ZZ 

945 

871 

.3009 

282 

27 

34 

968 

918 

.3001 

264 

26 

35 

.60991 

.76964 

1.2993 

.79247 

25 

36 

.61015 

.77010 

.2985 

229 

24 

37 

038 

057 

.2977 

211 

23 

38 

061 

103 

.2970 

193 

22 

39 
40 

084 

149 

.2962 

176 

21 
20 
19 

.61107 

.77196 

1.2954 

.79158 

41 

130 

242 

.2946 

140 

42 

153 

289 

.2938 

122 

18 

43 

176 

335 

.2931 

105 

17 

44 

199 

382 

.2923 

087 

16 

45 

.61222 

.77428 

1.2915 

.79069 

15 

46 

245 

475 

.2907 

051 

14 

47 

268 

521 

.2900 

033 

13 

48 

291 

568 

.2892 

.79016 

12 

49 

314 

615 

.2884 

.78998 

11 

50 

51 

.61337 

.77661 

1.2876 
.2869 

.78980 

10 

360 

708 

962 

9 

52 

Z%Z 

754 

.2861 

944 

8 

53 

406 

801 

.2853 

926 

7 

54 

429 

848 

.2846 

908 

6 

55 

.61451 

.77895 

1.2838 

.78891 

5 

56 

474 

941 

.2830 

873 

4 

57 

497 

.77988 

.2822 

855- 

3 

58 

520 

.78035 

.2815 

837 

2 

59 

543 

082 

.2807 

819 

1 
0 

60 

.61566 

.78129 

1.2799 

.78801 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

38° 


189 


1 
/ 

1 

2 
3 

N.  Sin. 

N.Tan. 

N.  Cot.  N.  Cos. 

.61566 

.78129 

1.2799 

.78801 

60 

589 
612 
635 

175 
222 
269 

.2792 

.2784 
.2776 

783 
765 

747 

59 

58 
57 

4 
5 
6 

658 

.61681 

704 

316 

.78363 

410 

.2769 

1.2761 

.2753 

729 

.78711 

694 

56 
55 
54 

7 

8 

9 

10 

726 
749 

772 

457 
504 
551 

.2746 
.2738 
.2731 

676 
658 
640 

53 
52 
51 
50 

.61795 

.78598 

1.2723 

.78622 

11 
12 
13 

818 
841 
864 

645 
692 
739 

.2715 
.2708 
.2700 

604 

586 
568 

49 

48 

47 

14 
15 
16 

887 

.61909 

932 

786 

.78834 

881 

.2693 

1.2685 

.2677 

550 

.78532 

514 

46 
45 

44 

17 
18 
19 

955 
.61978 
.62001 

928 
.78975 
.79022 

.2670 
.2662 
.2655 

496 

478 
460 

43 

42 
41 

20 

.62024 

.79070 

1.2647 

.78442 

40 

39 

38 
37 

21 
22 
23 

046 
069 
092 

117 
164 
212 

.2640 
.2632 
.2624 

424 
405 
387 

24 
25 
26 

115 

.62138 

160 

259 

.79306 

354 

.2617 

1.2609 

.2602 

369 

.78351 
333 

36 

35  S 

34  1 

27 
28 
29 

183 
206 
229 

401 
449 
496 

.2594 
.2587 
.2579 

315 
297 
279 

33] 
32  \ 
31  1 

30 

.62251 

.79544 

1.2572 

.78261 

30  i 

31 
32 
33 

274 
297 
320 

591 
639 
686 

.2564 
.2557 
.2549 

243 
225 
206 

29 

28 
27 

34 
35 
36 

342 

.62365 

388 

734 

.79781 

829 

.2542 

1.2534 

.2527 

188 

.78170 

152 

26 
25 
24 

37 
38 
39 

411 

433 
456 

877 

924 

.79972 

.2519 
.2512 
.2504 

134 
116 
098 

23 
22 
21 
20 
19 
18 
17 

40 

.62479 

.80020 

1.2497 

.78079 

41 
42 
43 

502 
524 
547 

067 
115 
163 

.2489 
.2482 
.2475 

061 
043 
025 

44 

45 
46 

570 

.62592 

615 

211 

.80258 

306 

.2467 
1.2460 

.2452 

.78007 

.77988 

970 

16 
15 
14 

47 
48 
49 

638 
660 

683 

354 
402 
450 

.2445 
.2437 
.2430 

952 
934 
916 

13 
12 
11 
10 

50 

51 
52 
53 

.62706 

.80498 

1.2423 

.77897 

728 
751 
774 

546 
594 
642 

.2415 
.2408 
.2401 

879 
861 
843 

9 

8 
7 

54 
55 
56 

796 

.62819 

842 

690 

.80738 
786 

.2393 

1.2386 

.2378 

824 
.77806 

788 

6 
5 
4 

57 
58 
59 
60 

864 
887 
909 

834 
882 
930 

.2371 
.2364 
.2356 

769 

751 
733 

3 
2 
1 
0 

.62932 

.80978 

1.2349 

1  .77715 

N.  Cos. 

N.  Cot.  N.Tan. 

|N.  Sin. 

/ 

30 


31 
32 
33 
34 
35 
36 
37 
38 
39 


40 


50^ 

51 

52 
53 
54 
55 
56 
57 
58 

60 


N.  Sin.  N.Tan.  N.  Cot.  N.  Cos. 


.62932 


955 

.62977 

.63000 

022 

.63045 

068 

090 

113 

135 


.63158 


180 
203 
225 
248 
.63271 
293 
316 
338 
361 


.63608 


630 
653 
675 
698 
.63720 
742 
765 
787 
810 


.63832 


854 

877 

899 

922 

.63944 

966 

.63989 

.64011 

033 


.64056 


078 
100 
123 
145 
.64167 
190 
212 
234 
256 


.64279 


N.  Cos 


.80978 


.81027 
075 
123 
171 

.81220 
268 
316 
364 
413 


.81946 


.81995 
.82044 
092 
141 
.82190 
238 
287 
336 
385 


.82434 


483 
531 
580 
629 

.82678 
727 
776 
825 
874 


.82923 


.82972 
.83022 
071 
120 
.83169 
218 
268 
317 
366 


.83415 


465 
514 
564 
613 
.83662 
712 
761 
811 
860 


.83910 


N.  Cot. 


1.2349 


.2342 
.2334 
.2327 
.2320 
1.2312 
.2305 
.2298 
.2290 
.2283 


1.2276 


.2268 
.2261 
.2254 
.2247 
1.2239 
.2232 
.2225 
.2218 
.2210 


1.2203 


.2196 
.2189 
.2181 
.2174 
1.2167 
.2160 
.2153 
.2145 
.2138 


1.2131 


.2124 
.2117 
.2109 
.2102 
1.2095 
.2088 
.2081 
.2074 
.2066 


1.2059 


.2052 
.2045 
.2038 
.2031 
1.2024 
.2017 
.2009 
.2002 
.1995 


1.1988 


.1981 
.1974 
.1967 
.1960 
1.1953 
.1946 
.1939 
.1932 
.1925 


1.1918 


N.Tan. 


■77715 
696 
678 
660 
641 

77623 
605 
586 
568 
550 


77531 
513 
494 
476 
458 

.77439 
421 
402 
384 
366 


.77347 


329 
310 
292 
273 
.77255 
236 
218 
199 
181 


.77162 


144 

125 

107 

088 

.77070 

051 

033 

.77014 

.76996 

.76977 


959 
940 
921 
903 

.76884 
866 
847 
828 
810 


.76791 


772 
754 
735 
717 
.76698 
679 
661 
642 
623 


.76604 


N.  Sin, 


?;io 


51 


50° 


190 

40° 

1 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.64279 

.83910 

.83960 

.84009 

059 

1.1918 

.76604 

60 

59 

58 
57 

1 

2 
3 

301 
323 
346 

.1910 
.1903 
.1896 

586 
567 
548 

4 
5 
6 

368 

.64390 

412 

108 

.84158 

208 

.1889 

1.1882 

.1875 

530 

.76511 

492 

56 
55 
54 

7 
8 
9 

435 
457 
479 

258 
307 
357 

.1868 
.1861 
.1854 

473 
455 
436 

53 
52 
51 

10 

11 
12 
13 

.64501 

.84407 

1.1847 

.76417 

50 

524 
546 
568 

457 
507 
556 

.1840 
.1833 
.1826 

398 
380 
361 

49 
48 

47 

14 
15 
16 

590 

.64612 

635 

606 

.84656 

706 

.1819 

1.1812 

.1806 

342 

.76323 

304 

46 
45 
44 

17 
18 
19 

657 
679 
701 

756 
806 
856 

.1799 
.1792 

.1785 

286 
267 

248 

43 
42 
41 
40 

20 

.64723 

.84906 

1.1778 

.76229 

21 
22 
23 

746 
768 
790 

.84956 

.85006 

057 

.1771 
.1764 
.1757 

210 
192 
173 

39 

38 
37 

24 
25 
26 

812 

.64834 

856 

107 

.85157 

207 

.1750 

1.1743 

.1736 

154 

.76135 

116 

36 
35 
34 

27 
28 
29 

878 
901 
923 

257 
308 
358 

.1729 
.1722 
.1715 

097 
078 
059 

32 
31 
30 

29 

28 

27 

30 

.64945 

.85408 

1.1708 

.76041 

31 
32 
2,2> 

967 
.64989 
.65011 

458 
509 
559 

.1702 
.1695 
.1688 

022 
.76003 
.75984 

34 
35 
36 

033 

.65055 

077 

609 

.85660 

710 

.1681 

1.1674 

.1667 

965 

.75946 

927 

26 
25 
24 

37 
38 
39 

100 
122 

144 

761 
811 
862 

.1660 
.1653 
.1647 

908 
889 
870 

23 
22 
21 
20 
19 
18 
17 

40 

.65166 

.85912 

1.1640 

.75851 

41 
42 

43 

188 
210 

232 

.85963 

.86014 

064 

.1633 
.1626 
.1619 

832 
813 
794 

44 
45 
46 

254 

.65276 

298 

115 

.86166 

216 

.1612 

1.1606 

.1599 

775 

.75756 

738 

16 
15 
14 

47 
48 
49 

320 

342 
364 

267 
318 
368 

.1592 
.1585 
.1578 

719 
700 

680 

13 
12 
11 
10 

9 

8 

7 

50 

.65386 

.86419 

1.1571 

.75661 

51 

52 
53 

408 
430 
452 

470 
521 

572 

.1565 
.1558 
.1551 

642 
623 
604 

54 
55 
56 

474 

.65496 

518 

623 
.86674 

725 

.1544 

1.1538 

.1531 

585 
.75566 

547 

6 

5 
4 

57 
58 
59 
60 

540 
562 
584 

776 
827 
878 

.1524 
.1517 
.1510 

528 
509 
490 

3 
2 
1 
0 

/ 

.65606 

.86929 

1.1504 

.75471 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

41° 


'" 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.65606 

.86929 

1.1504 

.75471 

60 

1 

2 
3 

628 
650 
672 

.86980 

.87031 

082 

.1497 
.1490 
.1483 

452 
433 
414 

59 

58 

57 

4 
5 
6 

694 
.65716 

738 

133 

.87184 

236 

.1477 

1.1470 

.1463 

395 

.75375 
356 

56 

55 
54 

7 
8 
9 

759 

781 
803 

287 
338 
389 

.1456 
.1450 
.1443 

337 
318 
299 

53 
52 
51 

10 

.65825 

.87441 

1.1436 

.75280 

50 

11 
12 
13 

847 
869 
891 

492 
543 
595 

.1430 
.1423 
.1416 

261 
241 
222 

49 

48 
47 

14 
15 
16 

913 

.65935 

956 

646 

.87698 

749 

.1410 

1.1403 

.1396 

203 

.75184 

165 

46 

45 
44 

17 
18 
19 
20 

.65978 

.66000 

022 

801 
852 
904 

.1389 
.1383 
.1376 

146 
126 
107 

43 
42 
41 

.66044 

.87955 

1.1369 

.75088 

40 

21 
22 

23 

066 
088 
109 

.88007 
059 
110 

.1363 
.1356 
.1349 

069 
050 
030 

39 

38 
37 

24 
25 
26 

131 

.66153 

175 

162 

.88214 
265 

.1343 

1.1336 

.1329 

.75011 

.74992 

973 

36 
35 
34 

27 
28 
29 
30 

197 

218 
240 

317 
369 
421 

.1323 
.1316 
.1310 

953 
934 
915 

ZZ 
32 
31 
30 

.66262 

.88473 

1.1303 

.74896 

31 
32 
ZZ 

284 
306 
327 

524 
576 
628 

.1296 
.1290 
.1283 

876 
857 
838 

29 

28 
27 

34 
35 
36 

349 

.66371 

393 

680 
.88732 

784 

.1276 

1.1270 

.1263 

818 

.74799 

780 

26 
25 
24 

37 
38 
39 
40 

414 
436 
458 

836 
888 
940 

.1257 
.1250 
.1243 

760 
741 

722 

23 
22 
21 
20 

.66480 

.88992 

1.1237 

.74703 

41 

42 
43 

501 
523 
545 

.89045 
097 
149 

.1230 
.1224 
.1217 

683 
664 
644 

19 

18 
17 

44 
45 
46 

566 

.66588 

610 

201 

.89253 

306 

.1211 

1.1204 

.1197 

625 

.74606 

586 

16 
15 
14 

47 
48 
49 

632 
653 
675 

358 
410 
463 

.1191 

.1184 

.1178 

1.1171 

567 

548 
528 

13 
12 
11 
10 

50 

.66697 

.89515 

.74509 

51 

52 
53 

718 
740 
762 

567 
620 
672 

.1165 
.1158 
.1152 

489 
470 
451 

9 

8 
7 

54 
55 
56 

783 
.66805 

827 

725 

.89777 

830 

.1145 

1.1139 

.1132 

431 

.74412 

392 

6 

5 
4 

57 
58 
59 
60 

848 
870 
891 

883 

935 

.89988 

.1126 
.1119 
.1113 

373 
353 
334 

3 
2 
1 
0 

.66913 

.90040 

1.1106 

.74314 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

49' 


48° 


42° 

48° 

191 

, 

N.  Sin.  N.  Tan.  N.  Cot. 

N.  Cos. 

1   / 

"o" 

1 

2 
3 

N.  Sin- 

N.Tan. 

N.  Cot. 

N.  Cos. 

0 

.66913 

.90040 

1.1106 

.74314 

60  ! 

.68200 

.93252 

1.0724 

.73135 

60 

1 

2 
3 

935 
956 
978 

093 
146 
199 

.1100 
.1093 
.1087 

295 
276 
256 

59 

58 
57 

221 
242 
264 

306 
360 
415 

.0717 
.0711 
.0705 

116 
096 
076 

59 

58 
57 

4 
5 
6 

.66999 

.67021 

043 

251 

.90304 

357 

.1080 

1.1074 

.1067 

237 

.74217 

198 

56 
55 
54 

4 
5 
6 

285 

.68306 

327 

469 
.93524 

578 

.0699 

1.0692 

.0686 

056 
.73036 
.73016 

56 

55 
54 

7 
8 
9 

064 
086 
107 

410 
463 
516 

.1061 
.1054 
.1048 

178 
159 
139 

53 
52 
51 

7 
8 
9 

349 
370 
391 

633 

688 
742 

.0680 
.0674 
.0668 

.72996 
976 
957 

53 
52 
51 

10 

.67129 

.90569 

1.1041 

.74120 

50 

10 

.68412 

.93797 

1.0661 

.72937 

50 

11 
12 
13 

151 
172 
194 

621 
674 

727 

.1035 
.1028 
.1022 

100 
080 
061 

49 

48 
47 

11 
12 
13 

434 
455 
476 

852 

906 

.93961 

.0655 
.0649 
.0643 

917 
897 

877 

49 

48 
47 

14 
15 
16 

215 

.67237 

258 

781 

.90834 

887 

.1016 

1.1009 

.1003 

041 
.74022 
.74002 

46 

45 
44 

14 
15 
16 

497 

.68518 

539 

.94016 

.94071 

125 

.0637 

1.0630 

.0624 

857 

.72837 

817 

46 

45 
44 

17 
18 
19 
20 

280 
301 
323 

940 
.90993 
.91046 

.0996 
.0990 
.0983 

.73983 
963 
944 

43 
42 
41 

17 
18 
19 

561 
582 
603 

180 
235 
290 

.0618 
.0612 
.0606 

797 

777 
757 

43 
42 
41 

.67344 

.91099 

1.0977 

.73924 

40 

20 

.68624 

.94345 

1.0599 

.72737 

40 

21 

22 
23 

366 
387 
409 

153 
206 
259 

.0971 
.0964 
.0958 

904 
885 
865 

39 

38 
37 

21 
22 
23 

645 
666 
688 

400 
455 
510 

.0593 
.0587 
.0581. 

717 
697 
677 

39 

38 
37 

24 
25 
26 

430 

.67452 

473 

313 

.91366 

419 

.0951 

1.0945 

.0939 

846 

.73826 

806 

36 
35 
34 

24 
25 
26 

709 

.68730 

751 

565 

.94620 

676 

.0575 

1.0569 

.0562 

657 

.72637 

617 

36 

35 
34 

27 
28 
29 
30 

495 
516 

538 

473 

526 

580 

.91633 

.0932 
.0926 
.0919 

787 
767 

747 

33 
32 
31 

27 
28 
29 

772 
793 
814 

731 
786 
841 

.0556 
.0550 
.0544 

597 

577 
557 

33 
32 
31 

.67559 

1.0913 

.73728 

30 

30 

.68835 

.94896 

1.0538 

.72537 

30 

31 
32 
33 

580 
602 
623 

687 
740 
794 

.0907 
.0900 
.0894 

708 
688 
669 

29 

28 

27 

31 

32 
33 

857 
878 
899 

.94952 

.95007 

062 

.0532 
.0526 
.0519 

517 
497 

477 

29 

28 
27 

34 
35 
36 

645 

.67666 

688 

847 
.91901 
.91955 

.0888 

1.0881 

.0875 

649 

.73629 

610 

26 
25 
24 

34 
35 
36 

920 

.68941 

962 

118 

.95173 

229 

.0513 

1.0507 

.0501 

457 

.72437 

417 

26 

25 
24 

37 
38 
39 
40 

709 
730 

752 

.92008 
062 
116 

.0869 
.0862 
.0856 

590 
570 
551 

23 
22 
21 
20 

37 
38 
39 

.68983 

.69004 

025 

284 
340 
395 

.0495 
.0489 
.0483 

397 

377 
357 

23 
22 
21 

.67773 

.92170 

1.0850 

.73531 

40 

.69046 

.95451 

1.0477 

.72337 

20 

41 
42 
43 

795 
816 

837 

224 

277 
331 

.0843 
.0837 
.0831 

511 
491 

472 

19 
18 
17 

41 
42 
43 

067 
088 
109 

506 
562 
618 

.0470 
.0464 
.0458 

317 
297 

277 

19 
18 
17 

44 
45 
46 

859 

.67880 

901 

385 

.92439 

493 

.0824 

1.0818 

.0812 

452 

.73432 

413 

16 
15 
14 

44 
45 
46 

130 

.69151 

172 

673 
.95729 

785 

.0452 

1.0446 

.0440 

257 

.72236 

216 

16 
15 
14 

47 
48 
49 
50 

923 
944 
965 

547 

601 

655 

.92709 

.0805 
.0799 
.0793 

393 

373 
353 

13 
12 
11 

47 
48 
49 
50 

51 

52 
53 

193 

214 

235 

841 

897 

.95952 

.0434 
.0428 
.0422 

196 
176 
156 

13 
12 
11 
10 

.67987 

1.0786 

.73333 

10 

.69256 

.96008 

1.0416 

.72136 

51 

52 
53 

.68008 
029 
051 

763 
817 
872 

.0780 
.0774 
.0768 

314 
294 

274 

9 

8 

7 

277 
298 
319 

064 
120 
176 

.0410 
.0404 
.0398 

116 
095 
075 

9 

8 

7 

54 
55 
56 

072 

.68093 

115 

926 
.92980 
.93034 

.0761 

1.0755 

.0749 

254 

.73234 

215 

6 
5 
4 

54 

55 
56 

340 
.69361 

382 

232 

.96288 

344 

.0392 

1.0385 

.0379 

055 
.72035 
.72015 

6 
5 
4 

57 
58 
59 

136 
157 
179 

088 
143 
197 

.0742 
.0736 
.0730 

195 
175 
155 

3 
2 

1 

57 
58 
59 
60 

403 
424 
445 

400 

457 
513 

.0373 
.0367 
.0361 

.71995 
974 
954 

3 
2 
1 

60 

.68200 

.93252 

1.0724 

.73135 

0 

.69466 

.96569 

1.0355 

.71934 

0 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

r 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

1 

47' 


ifV 


192 


44< 


/ 

N.  Sin. 

N.Tan. 

N.  Cot. 

N.  Cos. 

60 

59 

0 

1 

.69466 

.96569 

1.0355 

.71934 

487 

625 

.0349 

914 

2 

508 

681 

.0343 

894 

58 

3 

529 

738 

.0337 

873 

57 

4 

549 

794 

.0331 

853 

56 

5 

.69570 

.96850 

1.0325 

.71833 

55 

6 

591 

907 

.0319 

813 

54 

7 

612 

.96963 

.0313 

792 

53 

8 

633 

.97020 

.0307 

772 

52 

9 
10 

654 

076 

.0301 

752 
.71732 

51 
50 

.69675 

.97133 

1.0295 

11 

696 

189 

.0289 

711 

49 

12 

717 

246 

.0283 

691 

48 

13 

737 

302 

.0277 

671 

47 

14 

758 

359 

.0271 

650 

46 

15 

.69779 

.97416 

1.0265 

.71630 

45 

16 

800 

472 

.0259 

610 

44 

17 

821 

529 

.0253 

590 

43 

18 

842 

586 

.0247 

569 

42 

19 

862 

643 

.0241 

549 

41 
40 

20 

.69883 

.97700 

1.0235 

.71529 

21 

904 

756 

.0230 

508 

39 

22 

925 

813 

.0224 

488 

38 

23 

946 

870 

.0218 

468 

37 

24 

966 

927 

.0212 

447 

36 

25 

.69987 

.97984 

1.0206 

.71427 

35 

26 

.70008 

.98041 

.0200 

407 

34 

27 

029 

098 

.0194 

386 

33 

28 

049 

155 

.0188 

366 

32 

29 

070 

213 

.0182 

345 

31 
30 

30 

.70091 

.98270 

1.0176 

.71325 

31 

112 

327 

.0170 

305 

29 

32 

132 

384 

.0164 

284 

28 

33 

153 

441 

.0158 

264 

27 

34 

174 

499 

.0152 

243 

26 

35 

.70195 

.98556 

1.0147 

.71223 

25 

36 

215 

613 

.0141 

203 

24 

37 

236 

671 

.0135 

182 

23 

38 

257 

728 

.0129 

162 

22 

39 

277 

786 

.0123 

141 

21 

40 

.70298 

.98843 

1.0117 

.71121 

20 

41 

319 

901 

.0111 

100 

19 

42 

339 

.98958 

.0105 

080 

18 

43 

360 

.99016 

.0099 

059 

17 

44 

381 

073 

.0094 

039 

16 

45 

.70401 

.99131 

1.0088 

.71019 

15 

46 

422 

189 

.0082 

.70998 

14 

47 

443 

247 

.0076 

978 

13 

48 

463 

304 

.0070 

957 

12 

49 
50 

484 

362 

.0064 

937 

11 
10 

9 

.70505 

.99420 

1.0058 

.70916 

51 

525 

478 

.0052 

896 

52 

546 

536 

.0047 

875 

8 

53 

567 

594 

.0041 

855 

7 

54 

587 

652 

.0035 

834 

6 

55 

.70608 

.99710 

1.0029 

.70813 

5 

56 

628 

768 

.0023 

793 

4 

57 

649 

826 

.0017 

772 

3 

58 

670 

884 

.0012 

752 

2 

59 

690 

.99942 

.0006 

731 

1 
0 

60 

.70711 

1.0000 

1.0000 

.70711 

N.  Cos. 

N.  Cot. 

N.Tan. 

N.  Sin. 

r 

45° 


EXPLANATION  OF  TABLES. 

1.  Definition  of  logarithms.  The  logarithm  of  a  number 
to  the  base  k  is  defined  as  the  power  to  which  k  must  be  raised 
in  order  to  equal  the  number.  That  is,  if  the  logarithm  of  A 
to  the  base  k  is  a,  then  k"  =  A.  We  may  then  write  log;^  A  =  a. 
This  equation  is  read  "The  logarithm  of  A  to  the  base  k  equals  a." 

Since  2^'  =  S,  we  have  log2  8  =  3.  Similarly  since  2^  =  4  and 
32  =  9,  we  have  log2  4  =  2  and  logs  9  =  2. 

Example:  Find  the  values  of  logs  27,  logio  1000,  log2  32,  log^^  k, 

log,  1. 

2.  Logarithms  of  products,  quotients,  powers  and  roots. 

Let  A  =  k''  and  B  =  k*  so  that  log,  A  =  a  and  log,  B  =  b. 


Then 

AB  =  rk'  =  k'+K 

Hence 

log,  (AB)  =a  +  h. 

Therefore 

log*(AB)  =log,A+log,B. 

Also 

A  -^  B  =  k''  -^  k'  =  k'^. 

Hence 

log,  (A-i-  B)  =  a-h. 

Therefore 

log,  {A  -^  B)  =  log,  A  -  log,  B. 

And 

A"  =  (kT  =  k''^ 

Hence 

log,  (A")  =  na. 

Therefore 

log,  (A")  =  nlog,  A. 

And 

*^A  =  (k'')^=k^\ 

Hence 

log,  ^A  =  -a. 

Therefore 

log,  ^A  =-log,A. 

These  four  results  may  be  stated  as  follows: 
1.  The  logarithm  of  the  product  of  two  numbers  equals  the  sum 
of  the  logarithms  of  the  two  numbers. 
13  193 


194  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

2.  The  logarithm  of  the  quotient  of  two  numbers  equals  the 

logarithm  of  the  numerator  minus  the  logarithm  of  the 
denominator. 

3.  The  logarithm  of  the  nth.  power  of  a  number  equals  n  times 

the  logarithm  of  the  number. 

4.  The  logarithm  of  the  nth  root  of  a  number  equals  1/n  times 

the  logarithm  of  the  number. 

3.  Common*  Logarithms.  For  numerical  calculations  the 
most  convenient  system  of  logarithms  is  that  in  which  the  base 
is  10.  Logarithms  to  the  base  10  are  called  common  logarithms. 
In  all  that  follows  common  logarithms  will  be  used  and  logio  A 
will  be  denoted  simply  by  log  A,  etc. 

The  Characteristic.  Since  10^  =  10  and  10^  =  100,  we  have 
log  10  =  1  and  log  100  =  2,  and  hence  the  logarithm  of  any 
number  between  10  and  100  equals  a  number  between  1  and  2, 
and  is  therefore  made  up  of  1  plus  a  decimal.  The  whole 
number  1  is  called  the  characteristic  of  the  logarithm  and  the 
decimal  part  is  called  the  mantissa.  Similarly  the  logarithm  of  a 
number  between  100  and  1000  equals  2  plus  a  decimal,  2  being 
the  characteristic  and  the  decimal  the  mantissa  of  this  logarithm. 
Further,  since  10''  =  1  and  10^  =  10,  the  logarithm  of  a  number 
between  1  and  10  is  between  0  and  1  and  the  characteristic  of 
such  a  logarithm  is  then  0. 

Since  W  =  1  and  10~^  =  0.1,  the  logarithm  of  a  number 
between  0.1  and  1  is  between  —  1  and  0  and  may  therefore  be 
written  —  1  +  a  decimal.  The  characteristic  of  such  a  loga- 
rithm is  then  —  1.  The  mantissa  is  always  considered  positive 
whether  the  logarithm  is  a  positive  or  negative  number. 

Suppose  the  characteristic  of  a  logarithm  is  —  2  and  the 
mantissa  .56253.  This  logarithm  is  then  —  2  +  .56253  for 
which  we  use  the  notation  2.56253,  the  minus  sign  being  written 
above  the  characteristic  to  show  that  it  alone  is  negative.  In 
almost  all  numerical  calculations  it  is  found  to  be  more  con- 
venient to  add  and  subtract  10,  so  that  the  logarithm  takes  the 
form  8.56253  —  10.  In  practice  the  —  10  is  generally  left  off 
when  there  is  no  danger  of  confusion,  but  its  existence  must 
be  remembered. 

Example.     What  are  the  characteristics  of  the  logarithms  of 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  195 

the  following  numbers:  315.72,  7.6523,  0.354,  0.000673,  27.0054? 
The  Mantissa.  Consider  two  numbers  made  up  of  the  same 
sequence  of  digits  but  in  which  the  position  of  the  decimal  place 
is  different,  such  as  723.51  and  7.2351.  Now  723.51  =  10^  X 
7.2351.     Hence,  by  art.  2,  rule  1, 

log  723.51  =  log  102  +  log  7.2351. 

But  log  10^  =  2,  and  hence  . 

log  723.51  =  2  +  log  7.2351. 

That  is,  the  logarithms  of  these  two  numbers  differ  only 
in  the  characteristics,  and  the  mantissse  are  the  same. 
Obviously  two  numbers  made  up  of  the  same  sequence  of 
digits,  but  differing  in  the  position  of  the  decimal  point,  are 
such  that  either  is  equal  to  the  product  of  the  other  by  an  integral 
power  of  10,  and  hence  the  mantissae  of  the  logarithms  of  numbers 
differing  only  in  the  position  of  the  decimal  point  are  the  same. 
For  this  reason  in  tables  of  logarithms  the  mantissse  alone  are 
given,  the  characteristics  being  determined  by  the  two  rules 
below. 

From  the  foregoing  we  may  formulate  a  rule  for  finding  the 
characteristic  of  the  logarithm  of  any  number.  A  number  with 
five  digits  to  the  left  of  the  decimal  place,  such  as  13256.7,  is 
obviously  between  10,000  and  100,000,  that  is,  between  lO'*  and 
10^  Its  logarithm  is  then  between  4  and  5,  and  the  character- 
istic is  therefore  4. 

Similarly  the  characteristic  of  the  logarithm  of  a  number 
having  three  digits  to  the  left  of  the  decimal  place  is  2,  and 
finally  the  logarithm  of  a  number  having  n  digits  to  the  left 
of  the  decimal  place  is  n  —  1 

Hence,  the  characteristic  of  the  logarithm  of  a.  number  is  one 
less  than  the  number  of  digits  to  the  left  of  the  decimal  place. 

This  rule  obviously  does  not  apply  to  the  logarithms  of 
numbers  less  than  1.  Consider  the  logarithm  of  a  number 
having  no  integral  part  and  two  zeros  to  the  right  of  the  decimal 
before  the  first  significant  digit  appears,  such  as  0.00325.  This 
number  is  greater  than  one  one-thousandth  and  less  than  one 
one-hundredth  and  is  therefore  between  10~^  and   10"^.     Its 


196  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

logarithm  is  then  —  3  +  a  decimal,  so  that  the  characteristic 
is  —  3.     Similarly  the  characteristic  of  log  0.041  is  —  2. 

Hence  for  the  logarithms  of  numbers  less  than  1  the  character- 
istic is  minus  one  more  than  the  number  of  zeros  to  the  right  of 
the  decimal  place  before  the  first  significant  digit. 

Table  I. 

4.    To  find  the  Logarithm  of  a  Number. 

(a)  When  the  number  has  four  figures.  Find  the  first  three 
figures  of  the  number  on  pages  104-121  in  the  column  under  N 
and  the  fourth  figure  in  the  top  row.  The  last  three  figures  of 
the  mantissa  are  found  in  the  column  and  row  so  determined, 
and  the  first  two  figures  are  found  in  the  column  under  L.  The 
proper  characteristic  is  then  prefixed.  Thus  to  find  log  62.48 
we  look  for  624  under  N.  This  is  found  on  page  114,  and  the 
first  two  figures  of  the  mantissa  are  79.  The  last  three  figures 
of  the  mantissa  are  found  opposite  624  and  under  8.  They  are 
574.  Finally,  the  characteristic  is  1,  and  therefore  log  62.48 
=  1.79574. 

Whenever  the  three  last  figures  of  the  mantissa  are  preceded 

in  the  tables  by  an  asterisk  (*)  it  will  be  found  that  in  this  row 

78  ) 
under  L  two  numbers  are  bracketed,  as  «q  V  opposite  616  on 

page  114.  In  all  such  cases  the  upper  of  the  bracketed  numbers 
is  to  be  used  for  the  first  two  figures  of  the  mantissa  when  there 
is  not  an  asterisk,  and  the  lower  when  there  is  one.  Thus 
log  616.4  =  2.78986  and  log  616.7  =  2.79007.  Similarly  we 
find  log  0.7562  =  1.87864  or  9.87864-10;  log  0.02543  =  2.40535 
or  8.40535  -  10. 

(6)  When  the  number  has  three  or  less  figures.  Add  zeros  to 
the  right  of  the  number  and  then  find  the  logarithm  as  above. 
Thus  log  23  =  log  23.00  =  1.36173  (page  106),  etc. 

(c)  When  the  number  has  five  or  more  figures.  Let  us  be 
required  to  find  log  277.53.  Since  the  mantissa  does  not  depend 
upon  the  position  of  the  decimal  point,  we  may  find  it  by  finding 
the  mantissa  of  log  27753.  The  number  is  between  27750  and 
27760,  and  hence  its  logarithm  is  between  the  logarithms  of 
these  numbers.     It  is  therefore  equal  to  log  27750  +  x,  where 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  197 

a;  is  a  correction  to  be  added  to  log  27750.  We  assume  that  the 
difference  between  the  logarithms  is  proportional  to  the  differ- 
ence between  the  corresponding  numbers.  This  assumption  is 
only  approximately  true.     Now, 

the  mantissa  of  log  27753  =  .44326  +  x  ]  ^'«f  ^"^f  f  .'^"'^^^^^ 

log  27750  =  .44326  tJ'  ^""^  °[  ^°^'  =  ="' 

^      c^^^nr.        aa^ac,  [  difference  of  numbers 

log  27760  =  .44342  j      m      ^   a          ta 

*=  -'  =  10,  and  of  logs  =  16. 

We  then  assume  that      3  :  10  =  a:  :  16. 
And  hence  x  =  .3  X  16  =  4.8  =  5  - 

Therefore  log  277.53  =  2.44326  +  (5* 

=  2.44331 


16 

1.6 

3-f 
4-8 

6.4 
8.0 
9.6 

7  II. 2 

8  12.8 


The  multiplication  of  .3  by  16  is  most  easily  done        , 
by  aid  of  the  table  of  proportional  parts  found  in  the        ' 
logarithm  tables  to  the  right  of  the  logarithms.     Thus        J 
under  16  and  opposite  3  we  find  4.8  which  is  the  prod- 
uct of  16  by  .3.  11^.1 

From  the  foregoing  we  see  that  we  may  find  the  logarithm  of  a 
number  of  five  figures  as  follows. 

Find  the  mantissa  of  the  logarithm  of  the  first  four  figures,  and 
add  to  this  a  correction  found  hy  multiplying  the  difference  between 
this  mantissa  and  the  one  next  following  in  the  tables  {tabular  dif- 
ference) by  the  remaining  figure  of  the  given  number  considered 
as  a  decimal.  Finally  to  the  mantissa  so  obtained  prefix  the  proper 
characteristic. 

If  the  number  whose  logarithm  is  required  has  more  than 
five  figures  we  find  the  logarithm  of  the  nearest  number  of  five 
figures.  Thus,  for  log  380567  we  take  log  380570.  With 
logarithmic  tables  giving  only  five  places  of  decimals  there  is 
little  or  no  accuracy  gained  by  making  a  correction  for  a  sixth 
figure. 

In  the  correction  any  decimal  less  than  .5  is  neglected,  and  one 
over  .5  is  increased  to  1.  Thus  the  correction  for  log  18.202 
is  found  to  be  (page  105)  4.8,  and  this  is  taken  as  5  so  that 

log  18.202  =  1.26007 +  (5, 

*  The  symbol  ( is  written  before  the  correction  5  to  indicate  that  this  cor- 
rection is  to  be  added  to  the  last  figure  of  the  mantissa. 


198  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

the  5,  of  course,  being  added  to  the  last  figure  of  the  mantissa. 
Then  log  18.202=1.26012. 

When  a  correction  involves  exactly  .5  it  is  customary  to  either 
neglect  this  .5  or  to  increase  it  to  1  as  may  be  necessary  in  order 
to  make  the  last  figure  of  the  mantissa  an  even  number.  Thus 
log  19115  =  4.28126+ (11. 5.  We  then  take  the  correction  as  12 
so  that  log  19115=4.28138.  On  the  other  hand,  log  19105  = 
4.28103+ (11.5,  and  the  correction  is  here  taken  as  11,  so  that 
log  19105  =  4.28114.  This  arbitrary  rule  is  followed  as  such 
errors  are  more  apt  to  neutralize  one  another  than  would  be 
the  case  were  every  .5  to  be  either  neglected  or  increased 
to  1. 

5.  To  find  the  number  corresponding  to  a  given  logarithm. 
Let  us  be  required  to  find  the  number  n  when  given  that 
log  n=  1.08337.  The  given  mantissa  .08337  is  not  to  be  found 
in  the  tables,  but  it  lies  between  .08314  and  .08350.  Hence 
n,  apart  from  the  position  of  the  decimal  point,  lies  between 
1211  and  1212.  Therefore  n  =  1211+a;  where  a;  is  a  correction 
we  may  find  by  assuming  that  the  difference  between  two 
numbers  is  proportional  to  the  difference  between  their  loga- 
rithms. 

We  have,  then, 

i.-         r  1      1010     AoorA )  difference  of  logs  =  36,  and   of 
mantissa  of  log  1212  =  .08350  i        1  . 

1       1211  -  08*^14 -'^  ^^^^^^^' 

,  ""  \oooT  1  difference  of  logs  =  23,  and  of 

log  n      =  .08337  r        1 

°  )  numbers  =  a;. 

Therefore  a;:  1  =  23:  36. 
or  a;  =  23^36  =  .6+ 

Hence  the  sequence  of  numbers  making  up  n  is  12116. 

Finally,  since  the  logarithm  has  the  characteristic  1,  we  must 
point  off  two  places,  and  hence  w=  12.116. 

In  finding  the  correction  x  we  take  the  nearest  tenth  only. 
When  we  find  six  significant  figures  of  a  number  by  means  of 
its  logarithm  from  tables  giving  the  mantissae  to  five  places 
only,  the  sixth  figure  is  very  probably  incorrect,  and  any  attempt 
to  find  a  seventh  figure  would  be  of  absolutely  no  value. 

The  finding  of  the  correction  x  may  be  performed  more 
easily  by  means  of  the  tables  of  proportional  parts,  as  follows: 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  199 


36 

36 
7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
288 
9l32-4 


As  we  have  seen  a;  =  23-T-36.  That  is  x  is  the  quo-  , 
tient  obtained  by  dividing  the  difference  between  the  3 
given  mantissa  and  the  next  smaller  one  in  the  tables  \ 
by  the  tabular  difference  (the  difference  between  sue-  ^ 
cessive  mantissae  in  the  Tables). 

In  the  table  of  proportional  parts  under  the  tabular  differ- 
ence (36  in  this  case)  find  the  nearest  number  to  23.  This 
is  seen  to  be  21.6,  opposite  to  which  is  found  6.  We  then 
have  x  =  6. 

Examples. 

1.  Find  the  logarithms  of  25.897,  0.057281,  2537.3,  526.88. 

2.  Find  n  when  given  that  log  n  =  . 32380,  log  n  =  8.58720- 10. 

3.  Verify  the  equation  log  23.6+log  7.0004  =  log  (23.6X7.0004). 
1.  By  means  of  logarithms  find  to  five  figures  i/227.54. 

Table  II. 

6.  By  log  sin  d  we  mean  the  logarithm  of  that  number  which 
is  equal  to  sin  B,  and  similarly  for  the  logarithms  of  the  other 
functions  of  6.  In  Table  II  will  be  found  the  logarithms  of 
the  sines,  cosines,  tangents  and  cotangents  of  angles  from  0° 
to  90°,  computed  for  each  minute.  For  angles  from  0°  to  and 
including  44°  the  number  of  degrees  is  printed  at  the  top  of  the 
page  and  the  number  of  minutes  in  the  column  under  '  (to  the 
left  of  the  column  L.  Sin.) .  The  logarithms  of  the  sines  are 
found  under  the  heading  L.  Sin.,  those  of  the  tangents  under 
L.  Tan.,  etc.  For  angles  from  45°  to  90°  the  number  of  degrees 
is  printed  at  the  bottom  of  the  page,  the  number  of  minutes  is 
found  in  the  right  hand  column,  and  the  logarithms  of  the  sines 
in  the  column  over  L.  Sin.,  and  similarly  for  the  other  func- 
tions. Thus  on  page  147,  we  find  log  sin  23°  52'  =  9.60704, 
log  tan  23°  31'  =  9.63865,  log  sin  66°  18'  =  9.96174,  etc. 

Since  the  sine  and  cosine  of  any  angle  are  less  than  1,  the 
logarithms  of  these  functions  have  negative  characteristics. 
Also  since  the  tangent  of  an  angle  less  than  45°  is  less  than  1, 
the  logarithm  of  the  tangent  of  such  an  angle  has  a  negative 
characteristic.  All  of  the  logarithms  in  the  columns  under 
L.  Sin.,  L.  Tan.,  and  L.  Cos.,  are  too  great  by  10.     Thus, 


200  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

log  sin  23°  =  9.59188  —  10,  etc.  In  practice  it  is  customary  to 
leave  off  the  —  10  but  its  existence  must  be  remembered 
(art.  3).  The  logarithms  in  the  column  under  L.  Cot.  have 
correct  characteristics. 

7.  To  find  the  logarithm  of  the  sine  of  an  angle.  Let  us  be 
required  to  find  log  sin  42°  24'  35''.  The  required  logarithm 
will  be  between  log  sin  42°  24'  and  log  sin  42°  25'.  On  page  166 
we  find  log  sin  42°  24'  =  9.82885,  and  the  difference  between 
this  logarithm  and  log  sin  42°  25'  is  given  in  the  column  under 
d  as  14.  Then  the  required  logarithm  will  be  9.82885  +  x, 
where  x  is  a  correction  found  by  assuming  that  the  difference 
between  two  angles  is  proportional  to  the  difference  between 
the  logarithms  of  their  sines.     That  is,  a:  :  14  =  35  :  60,  or 

X  =  ^^^-^^  =  8.1+.     Hence  log  sin  42°  24'  35"  =  9.82885 

+  (8  =  9.82893. 

The  correction  to  be  added  to  9.82885  will  be  found  in  the 
table  of  proportional  parts  under  the  tabular  difference  (14) 
and  opposite  35".  With  a  little  practice  this  correction  may 
be  made  mentally  and  the  final  logarithm  alone  written  down. 

8.  To  find  the  logarithm  of  the  tangent  of  an  angle  the  method 
employed  is  exactly  similar  to  that  of  the  last  article.  Thus 
log  tan  56°  11'  45"  =  0.17401  +  (21  =  0.17422. 

9.  To  find  the  logarithms  of  the  cosine  and  cotangent  of  an  angle. 
Since  the  cosine  and  cotangent  of  angles  less  than  90°  decrease 
in  value  as  the  angles  increase  the  logarithms  of  these  functions 
also  decrease  as  the  angles  increase,  and  hence  the  correction 
for  the  given  number  of  seconds  must  be  subtracted.  Thus  to 
find  log  cos  57°  32'  25",  we  have,  page  156, 

log  cos  57°  32'  =  9.72982      (tabular  difference  =  20) 
correction  for  25"  =  8.3 

.-.  log  cos  57°  32'  25"  =  9.72974. 

Similarly, 

log  cot  33°  25'  =  0.18059     (tabular  difference  =  27) 
correction  for  20"  =  9 

.-.  log  cot  33°  25'  20"  =  0.18050. 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  201 

10.  To  find  the  angle  corresponding  to  the  logarithm  of  one  of 
its  functions.     Let  us  be  required  to  find  6  when  given 

log  sin  0  =  9.78350. 

On  page  161  we  find  that  the  given  logarithm  is  between 
log  sin  37°  24'  and  log  sin  37°  25'  (tabular  difference  = 
16).  Hence  d  =  37°  24'  +  x'\  The  difference  between  the 
given  logarithm  (9.78350)  and  log  sin  37°  24'  (=  9.78346)  is  4. 

Hence  x"  :  60"  =  4  :  16.      Then  x"  =^-t«^   =  15".     And 

lb 

therefore  9  =  37°  24'  15".     The  tables  of  proportional  parts 

may  be  used  to  find  the  number  of  seconds  as  follows:  in  the 

column  under  the  tabular  difference  find  the  number  nearest 

to  the  difference  between  the  given  logarithm  and  the  logarithm 

corresponding  to  the  next  smaller  angle  in  the  tables.     Opposite 

to  this  will  be  found  the  proper  number  of  seconds. 

Find  6  when  given  log  cos  6  =  9.80541.  On  page  163  we  find 
that  the  given  logarithm  is  between  log  cos  50°  17'  =  9.80550 
and  log  cos  50°  18'.  The  tabular  difference  is  16  and  the  dif- 
ference between  the  given  logarithm  and  that  corresponding  to 
the  smaller  of  these  two  angles  (50°  17')  is  9.  Hence  the  number 
of  seconds  is  found  by  looking  for  the  number  nearest  to  9  in 
the  table  of  proportional  parts  under  16.  This  is  9.3  and  corre- 
sponds to  35".     Hence  6  =  50°  17'  35". 

In  finding  the  number  of  seconds  we  take  the  nearest  number 
of  seconds  given  in  the  tables  of  proportional  parts  because  we 
are  not  sure  of  obtaining  accurate  results  closer  than  five 
seconds  with  tables  giving  only  five  places  of  decimals. 

Examples. 

1.  Find  log  sin  63°  13'  15",  log  cos  37°  42'  55",  log  cot  62°  22'  5", 

log  tan  21°  14'  40",  log  cot  14°  12'  50". 

2.  Find  d  when  given  log  sin  d  =  9.88645,  log  cos  6  =  9.32253, 

log   tan  e  =  0.25436,    log   cot  d  =  9.34152,    log   tan  6  = 
9.22335. 

3.  Given  a  =  24.762  X  tan  36°  42'  45",  find  a. 

4.  Given  a  =  2.0034  ycos  32°  24'  25",  find  a. 

5.  Find  the  numerical  values  of  cos^  37°  23'  5"  and  sin^  37°  23'  5", 

and  show  that  their  sum  is  unity.     (Owing  to  the  fact  that 


202  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

the  logarithmic  tables  give  only  five  places  of  decimals, 
there  may  be  an  error  in  the  last  decimal.) 
11.  To  find  the  logarithms  of  the  sine  and  tangent  of  angles  near 
0°.  The  method  we  have  employed  for  correcting  the  logarith- 
mic sines  and  tangents  for  seconds  does  not  give  reliable  resuli% 
for  angles  between  0°  and  4°.  To  find  such  logarithms  we 
make  use  of  two  quantities  S  and  T  defined  by  the  equations 


o       ,     sin  ^      .        .    ^       ,      ^,, 
>S  =  log-^  =  log  sm  e  -  log  61", 

^  =  log^7^  =  logtan0-logC 


(1) 


where  0"  is  the  number  of  seconds  in  6.    So  that 

log  smd  =  S  +  log  e", 
and 

log  tan  ^  =  r  +  log  e'\  (2) 

On  pages  124  to  128  will  be  found  the  proper  values  of  S  and  T, 
the  first  three  figures  being  printed  at  the  top  of  the  column  and 
the  last  three  opposite  the  required  angle.  The  characteristics 
of  S  and  T  in  the  tables  should  be  decreased  by  10.  Also  in  the 
column  headed  ''  will  be  found  the  number  of  seconds  in  each 
angle  given  in  the  tables. 

Let  us  be  required  to  find  log  sin  2°  32'  25". 

On  page  126  we  find  that  the  number  of  seconds  in  2°  32'  = 
9120,  and  hence  0  =  2°  32'  25"  =  9120"  +  25"  =  9145". 

Hence  we  have  S  =  4.68543  -  10 

log  e"  =  log  9145  =  3.96118  (page  120) 
.-.  log  sin  2°  32'  25"  =  8.64661  -  10. 

Similarly  we  may  find  log  tan  0°  26'  35"  as  follows. 

26'  35"  =  1560"  +  35"  =  1595". 

Hence    we    have  T  =  4.68558  -  10 

and  log  1595  =  3.20276 

.-.  log  tan  0°  26'  35"  =  7.88834  -  10. 

12.  To  find  the  logarithm  of  the  cotangent  of  an  angle  near  0°. 

Since  cot  0  =  -, -,  we  have  log  cotO  =  —  log  tan  6.    We  may 

tan  6 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  203 

then  find  log  tan  d  by  the  method  of  the  last  article  and,  upon 
changing  its  sign,  obtain  the  required  logarithm.  Thus,  in 
article  11  we  have  found  that  log  tan  0°  26'  35"  =  7.88834  -  10. 
Hence  we  have  log  cot  0°  26'  35"  =  10  -  7.88834  =  2.11166. 

13.  To  find  an  angle  between  0°  and  5°  when  given  the  logarithm 
of  its  sine,  tangent  or  cotangent. 

Let  us  be  required  to  find  6  when  given  log  sin  d  =  8.34106. 
On  page  125  we  find  that  6  is  between  1°  15'  and  1°  16",  and  that 
the  corresponding  value  of  S  is  4.68554  —  10.  Then,  since 
*S  =  log  sin  6  —  log  e,  we  have  log  d  =  log  sin  6  —  S. 

But  log  sin  e  =  8.34106  -  10 

S  =  4.68554  -  10 
log  0"  =  3.65552 

and  0"  =  4524"  =  1°  15'  24". 

In  a  similar  manner  we  may  find  6  when  given  log  tan  0.  Let 
log  tan  0  =  8.44932  -  10.  Then  d  is  between  1°  36'  and  1°  37' 
and   the   corresponding  value   of   T  is   4.68569  —  10.     Then, 

log  tan  d  =  8.44932  -  10 
T  =  4.68569  -  10 
therefore  log  0"  =  3.76363 

and  0"  =  5803"  -  =  1°  36'  43"  -. 

As  a  third  example  let  us  find  6  when  log  cot  0  =  1.63442. 
Then  as  in  article  12,  log  tan  0  =  —  log  cot  ^  =  —  1.63442  = 
8.36558  -  10.     Hence  d  is  between  1°  19'  and  1°  20',  and  the 
corresponding  value  of  T  is  4,68565  —  10. 
Then 

log  tan  d  =  8.36558  -  10 
T  =  4.68565  -  10 
therefore  log  0"  =  3.67993 

and  •       0"  =  4786  -  =  1°  19'  46"  -. 

13.  The  ordinary  method  for  finding  the  correction  for  seconds 
also  fails  when  we  attempt  to  find  the  logarithms  of  the  tan- 
gent, cotangent  and  cosine  of  angles  near  90°.  Now,  tan  0  = 
cot  (90°  —  0),  and  hence  we  may  find  log  tan  0  by  finding 
log  cot  (90°  -  0).     But  when  0  is  near  90°,  90°  -  0  is  near  0°, 


204  ELEMENTS  OF  PLANE  TRIGONOMETRY. 

and  therefore  log  cot  (90°  —  6)  may  be  found  by  the  method  of 
article  12.  In  like  manner  we  may  find  log  cos  6  by  finding 
log  sin  (90°  -  d),  and  log  cot  d  by  log  tan  (90°  -  6>). 

Thus  log  cot  89°  44'  45''  =  log  tan  0°  15'  15"  =  log  tan  915". 
Then  T  =  4.68558  -  10 

log  915  =  2.96142 
log  cot  89°  44'  45"  =  7.64700  -  10. 
As  a  second  problem  let  us  find  6  when  given  log  cos  6  = 
8.24104  -  10.     Then  we  have  log  sin  (90°  -  d)  =  8.24104  -  10. 
Hence  90°  -6  is  between  0°  59'  and  1°  0',  and  the  correspond- 
ing value  of  S  is  4.68555  -  10. 
Then  log  sin  (90°  -  d)  =  8.24104  -  10 

S  =  4.68555  -  10 
therefore  log  (90°  -  61)"  =  3.55549 

and  (90°  -  ^)"  =  3593"  +  =  59'  53". 

Hence  0  =  90°  -  59'  53"  =  89°  0'  7". 

14.  To  find  the  logarithms  of  functions  of  angles  not  in  the  first 
quadrant.  We  may  find  log  sin  120°  by  making  use  of  the  relation 
sin  120°  =  sin  (180°  -  120°),  and  looking  up  log  sin  60°. 
The  case  is,  however,  somewhat  different  if  we  need  to 
find  log  cos  120°,  for  cos  120°  is  a  negative  number  and  a 
negative  number  has  no  real  logarithm.  We  avoid  this  dif- 
ficulty as  follows:  we  have  cos  120°  =  —  cos  60°,  and  we  use 
log  cos  60°.  Since  logarithms  are  in  general  only  used  in 
problems  involving  multiplication  and  division,  we  may  con- 
sider any  product  or  quotient  as  made  up  only  of  positive 
quantities,  and  after  all  the  logarithmic  work  is  completed,  give 
the  result  the  proper  sign,  positive  when  an  even  number  of 
negative  quantities  are  involved  and  negative  when  this  number 
is  odd.  Even  though  A^  be  a  negative  number  it  is  customary  to 
write  log  N,  this  symbol  being  used  instead  of  the  strictly  proper 
log  {- N).  Thus  we  write  log  cos  120°  =  log  cos  60°  = 
9.69897. 

Examples. 
1.  Find  log  cos  132°  45'  35",  log  sin  2°  22'  15",  log  tan  1°  21'  55"; 
log  cot  0°  1'  5",  log  cos  87°  59'  30",  log  cot  88°  53'  50". 


ELEMENTS  OF  PLANE  TRIGONOMETRY.  205 

2.  Find  the  numerical  value  of  a  =  312.87  X  sin  2°  2'  20"  X 

cos  100°  43'  45''. 

3.  By  means   of  logarithms  verify  sin   140°   26'   40"  =  2  X 

sin  70°  13'  20"  X  cos  70°  13'  20". 

4.  Verify  the  relation  cos  2x  =  2  cos  (45°  —  x)  sin  (45°  --  a;), 

when  X  =  10°  13'  15" 

Table  III. 

15.  In  this  table  are  found  the  numerical  values  of  the  trigo- 
nometric functions  of  angles  between  0°  and  90°.  The  method 
of  using  this  table  is  exactly  similar  to  that  made  use  of  in 
finding  the  logarithms  of  the  trigonometric  functions  in  Table  II. 
Tables  of  proportional  parts  are,  however,  not  given,  and  so 
any  necessary  corrections  must  be  performed  by  actual  multipli- 
cation or  division. 

To  find  the  value  of  sin  36°  24'  25"  we  proceed  as  follows: 
From  page  88  we  find 

sin  36°  24'  =  .59342 
and  sin  36°  25'  =  .59365 

The  difference  between  these  numbers  is  23,  and  hence  the 
correction  to  be  added  to  sin  36°  24'  is       Q^       =  9.5  +.    The 

correction  is  then  taken  as  10  and  we  have  sin  36°  24'  25"  = 
.59342  +  (10  =  .59352. 
Examples. 

1.  Find  the  values  of 

tan  65°  51'  50",    cos  32°  12'  15", 
cot  14°  16'  55",    cos  154°  24'  25". 

2.  Find  sin  55°  24'  35".     Then  look  up  the  logarithm  of  this 

number  in  table  I,  and  show  that  the  result  so  obtained 
is  the  same  as  log  sin  55°  24'  35"  as  found  from  Table  II. 


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TRIGONOMETRY 

By  DAVID  A.  ROTHROCK,  Ph.D. 

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number  of  problems  illustrative  of  the  principles  demonstrated.  Answers 
to  the  problems  are  included. 


COLLEGE  MATHEMATICS— Cbn/mae^/ 


COORDINATE  GEOMETRY 

Analytic  Geometry  for  Technical  Schools  and  Colleges* 
By  P.  A.  Lambert,  Assistant  Professor  of  Mathematics, 
Lehigh  University. 

Qoih*  t2mo»  216  pages*  $t,50  net* 
The  object  is  to  furnish  a  natural  but  thorough  introduction  to 
the  principles  and  applications  of  analytical  geometry  with  a 
view  to  the  use  made  of  the  subject  by  engineers.  The  impor- 
tant engineering  curves  are  thoroughly  discussed  and  the  applica. 
tion  of  analytic  geometry  to  mathematics  and  physics  is  made  a 
special  point. 

Conic  Sections*  By  Charles  Smith,  Master  of  Sidney  Sus- 
sex College,  Cambridge. 

Cloth*  t2mo*  352  pages »  $f*60net* 
This  well-known  text,  which  has  been  reprinted  eighteen  times 
since  the  second  edition  was  issued  in  1833,  is  still  considered  a 
standard.  The  elementary  properties  of  the  straight  lines,  circle, 
quadrille,  ellipse,  and  hyperbola  are  discussed,  accompanied  by 
many  examples  selected  and  arranged  to  illustrate  principles. 


CALCULUS  AND  DIFFERENTIAL 

EQUATIONS 

A  First  Course  in  the  Differential  and  Integral  Calculus^ 
By  William  F.  Osgood,  Professor  of  Mathematics  in 
Harvard  University.  Qoth*     t2mo.    $2M  net. 

Designed  as  a  text  for  students  beginning  the  study  and  devoting 
to  it  about  one  year's  work.  The  principal  features  of  the  book 
are  the  introduction  of  the  integral  calculus  at  an  early  date ;  the 
introduction  of  the  practical  applications  of  the  subject  in  the 
first  chapters;  and  the  introduction  of  many  practical  problems 
of  engineering,  physics,  and  geometry.  The  problems,  over  900 
in  number,  have  been  chosen  with  a  view  to  presenting  the  ap- 
plications of  the  subject  not  only  to  geometry,  but  also  to  the 
practical  problems  of  physics  and  engineering. 


COLLEGE  MATHEMATICS— Cbn/mae(/ 


The  Elements  of  the  Differential  and  Integral  Calculus^ 
By  Donald  Francis  Campbell,  Professor  of  Mathe- 
matics, Armour  Institute  of  Technology. 

Cloth,  tlmo,  362  pages,  $t*90  net 

Written  to  meet  the  need  of  students  desiring  a  thoroughgoing  practical 
treatise  on  the  subject.  Il  is  primarily  designed  for  use  in  technical 
schools,  but  has  given  excellent  satisfaction  in  connection  with  general 
courses  in  classical  colleges  and  universities.  In  order  to  enable  the 
studeiit  to  grasp  more  fully  the  details  of  the  subject  the  a*uthor  has  intro- 
duced a  large  number  of  practical  questions  which  are  found  in  actual 
experience  to  produce  the  desired  result  better  than  theoretical  proposi- 
tions. 

Differential  and  Integral  Calculus  for  Technical  Schools 
and  Colleges*    By  P.  A.  Lambert. 

Qoih,  245  pages,  $t^50  net 

A  text  for  students  who  have  a  working  knowledge  of  elementary  geom- 
etry, algebra,  trigonometry,  and  analytical  geometry.  Its  object  is  three- 
fold :  (1 )  to  inspire  confidence  in  the  methods  of  infinitesimal  analysis  ; 
(2)  to  aid  in  acquiring  facility  in  applying  these  methods-,  and  (3)  to 
show  the  practical  value  of  the  calculus  by  applications  to  problems  in 
physics  and  engineering. 

An  Elementary  Treatise  on  the  Calculus*  By  George 
A.  Gibson,  Professor  of  Mathematics  in  Glasgow  and 
West  of  Scotland  Technical  College. 

Cloth,  \2mo,  5t8  pages,  $1.90  net 

A  text  for  college  classes  which  will  devote  a  year's  study  of  approxi- 
mately three  hours  per  week  to  the  subject.  The  treatment  has  been  as 
thorough  and  as  rigid  as  it  is  possible  to  give  in  an  elementary  treatise. 

An  Introduction  to  the  Calculus*   By  George  A.  Gibson. 

Cloth,  t2mo,  222  pages,  $0.90 

The  elements  of  calculus  have  been  treated  in  a  way  easily  understood  by 
immature  students.  The  reason  is  based  essentially  on  the  graphical 
representation  of  a  function. 


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